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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    <a name="line.17"></a>
<FONT color="green">018</FONT>    package org.apache.commons.math.util;<a name="line.18"></a>
<FONT color="green">019</FONT>    <a name="line.19"></a>
<FONT color="green">020</FONT>    import java.math.BigDecimal;<a name="line.20"></a>
<FONT color="green">021</FONT>    import java.math.BigInteger;<a name="line.21"></a>
<FONT color="green">022</FONT>    import java.util.Arrays;<a name="line.22"></a>
<FONT color="green">023</FONT>    <a name="line.23"></a>
<FONT color="green">024</FONT>    import org.apache.commons.math.MathRuntimeException;<a name="line.24"></a>
<FONT color="green">025</FONT>    <a name="line.25"></a>
<FONT color="green">026</FONT>    /**<a name="line.26"></a>
<FONT color="green">027</FONT>     * Some useful additions to the built-in functions in {@link Math}.<a name="line.27"></a>
<FONT color="green">028</FONT>     * @version $Revision: 772119 $ $Date: 2009-05-06 05:43:28 -0400 (Wed, 06 May 2009) $<a name="line.28"></a>
<FONT color="green">029</FONT>     */<a name="line.29"></a>
<FONT color="green">030</FONT>    public final class MathUtils {<a name="line.30"></a>
<FONT color="green">031</FONT>    <a name="line.31"></a>
<FONT color="green">032</FONT>        /** Smallest positive number such that 1 - EPSILON is not numerically equal to 1. */<a name="line.32"></a>
<FONT color="green">033</FONT>        public static final double EPSILON = 0x1.0p-53;<a name="line.33"></a>
<FONT color="green">034</FONT>    <a name="line.34"></a>
<FONT color="green">035</FONT>        /** Safe minimum, such that 1 / SAFE_MIN does not overflow.<a name="line.35"></a>
<FONT color="green">036</FONT>         * &lt;p&gt;In IEEE 754 arithmetic, this is also the smallest normalized<a name="line.36"></a>
<FONT color="green">037</FONT>         * number 2&lt;sup&gt;-1022&lt;/sup&gt;.&lt;/p&gt;<a name="line.37"></a>
<FONT color="green">038</FONT>         */<a name="line.38"></a>
<FONT color="green">039</FONT>        public static final double SAFE_MIN = 0x1.0p-1022;<a name="line.39"></a>
<FONT color="green">040</FONT>    <a name="line.40"></a>
<FONT color="green">041</FONT>        /** -1.0 cast as a byte. */<a name="line.41"></a>
<FONT color="green">042</FONT>        private static final byte  NB = (byte)-1;<a name="line.42"></a>
<FONT color="green">043</FONT>    <a name="line.43"></a>
<FONT color="green">044</FONT>        /** -1.0 cast as a short. */<a name="line.44"></a>
<FONT color="green">045</FONT>        private static final short NS = (short)-1;<a name="line.45"></a>
<FONT color="green">046</FONT>    <a name="line.46"></a>
<FONT color="green">047</FONT>        /** 1.0 cast as a byte. */<a name="line.47"></a>
<FONT color="green">048</FONT>        private static final byte  PB = (byte)1;<a name="line.48"></a>
<FONT color="green">049</FONT>    <a name="line.49"></a>
<FONT color="green">050</FONT>        /** 1.0 cast as a short. */<a name="line.50"></a>
<FONT color="green">051</FONT>        private static final short PS = (short)1;<a name="line.51"></a>
<FONT color="green">052</FONT>    <a name="line.52"></a>
<FONT color="green">053</FONT>        /** 0.0 cast as a byte. */<a name="line.53"></a>
<FONT color="green">054</FONT>        private static final byte  ZB = (byte)0;<a name="line.54"></a>
<FONT color="green">055</FONT>    <a name="line.55"></a>
<FONT color="green">056</FONT>        /** 0.0 cast as a short. */<a name="line.56"></a>
<FONT color="green">057</FONT>        private static final short ZS = (short)0;<a name="line.57"></a>
<FONT color="green">058</FONT>    <a name="line.58"></a>
<FONT color="green">059</FONT>        /** 2 &amp;pi;. */<a name="line.59"></a>
<FONT color="green">060</FONT>        private static final double TWO_PI = 2 * Math.PI;<a name="line.60"></a>
<FONT color="green">061</FONT>    <a name="line.61"></a>
<FONT color="green">062</FONT>        /** Gap between NaN and regular numbers. */<a name="line.62"></a>
<FONT color="green">063</FONT>        private static final int NAN_GAP = 4 * 1024 * 1024;<a name="line.63"></a>
<FONT color="green">064</FONT>    <a name="line.64"></a>
<FONT color="green">065</FONT>        /** Offset to order signed double numbers lexicographically. */<a name="line.65"></a>
<FONT color="green">066</FONT>        private static final long SGN_MASK = 0x8000000000000000L;<a name="line.66"></a>
<FONT color="green">067</FONT>    <a name="line.67"></a>
<FONT color="green">068</FONT>        /**<a name="line.68"></a>
<FONT color="green">069</FONT>         * Private Constructor<a name="line.69"></a>
<FONT color="green">070</FONT>         */<a name="line.70"></a>
<FONT color="green">071</FONT>        private MathUtils() {<a name="line.71"></a>
<FONT color="green">072</FONT>            super();<a name="line.72"></a>
<FONT color="green">073</FONT>        }<a name="line.73"></a>
<FONT color="green">074</FONT>    <a name="line.74"></a>
<FONT color="green">075</FONT>        /**<a name="line.75"></a>
<FONT color="green">076</FONT>         * Add two integers, checking for overflow.<a name="line.76"></a>
<FONT color="green">077</FONT>         * <a name="line.77"></a>
<FONT color="green">078</FONT>         * @param x an addend<a name="line.78"></a>
<FONT color="green">079</FONT>         * @param y an addend<a name="line.79"></a>
<FONT color="green">080</FONT>         * @return the sum &lt;code&gt;x+y&lt;/code&gt;<a name="line.80"></a>
<FONT color="green">081</FONT>         * @throws ArithmeticException if the result can not be represented as an<a name="line.81"></a>
<FONT color="green">082</FONT>         *         int<a name="line.82"></a>
<FONT color="green">083</FONT>         * @since 1.1<a name="line.83"></a>
<FONT color="green">084</FONT>         */<a name="line.84"></a>
<FONT color="green">085</FONT>        public static int addAndCheck(int x, int y) {<a name="line.85"></a>
<FONT color="green">086</FONT>            long s = (long)x + (long)y;<a name="line.86"></a>
<FONT color="green">087</FONT>            if (s &lt; Integer.MIN_VALUE || s &gt; Integer.MAX_VALUE) {<a name="line.87"></a>
<FONT color="green">088</FONT>                throw new ArithmeticException("overflow: add");<a name="line.88"></a>
<FONT color="green">089</FONT>            }<a name="line.89"></a>
<FONT color="green">090</FONT>            return (int)s;<a name="line.90"></a>
<FONT color="green">091</FONT>        }<a name="line.91"></a>
<FONT color="green">092</FONT>    <a name="line.92"></a>
<FONT color="green">093</FONT>        /**<a name="line.93"></a>
<FONT color="green">094</FONT>         * Add two long integers, checking for overflow.<a name="line.94"></a>
<FONT color="green">095</FONT>         * <a name="line.95"></a>
<FONT color="green">096</FONT>         * @param a an addend<a name="line.96"></a>
<FONT color="green">097</FONT>         * @param b an addend<a name="line.97"></a>
<FONT color="green">098</FONT>         * @return the sum &lt;code&gt;a+b&lt;/code&gt;<a name="line.98"></a>
<FONT color="green">099</FONT>         * @throws ArithmeticException if the result can not be represented as an<a name="line.99"></a>
<FONT color="green">100</FONT>         *         long<a name="line.100"></a>
<FONT color="green">101</FONT>         * @since 1.2<a name="line.101"></a>
<FONT color="green">102</FONT>         */<a name="line.102"></a>
<FONT color="green">103</FONT>        public static long addAndCheck(long a, long b) {<a name="line.103"></a>
<FONT color="green">104</FONT>            return addAndCheck(a, b, "overflow: add");<a name="line.104"></a>
<FONT color="green">105</FONT>        }<a name="line.105"></a>
<FONT color="green">106</FONT>        <a name="line.106"></a>
<FONT color="green">107</FONT>        /**<a name="line.107"></a>
<FONT color="green">108</FONT>         * Add two long integers, checking for overflow.<a name="line.108"></a>
<FONT color="green">109</FONT>         * <a name="line.109"></a>
<FONT color="green">110</FONT>         * @param a an addend<a name="line.110"></a>
<FONT color="green">111</FONT>         * @param b an addend<a name="line.111"></a>
<FONT color="green">112</FONT>         * @param msg the message to use for any thrown exception.<a name="line.112"></a>
<FONT color="green">113</FONT>         * @return the sum &lt;code&gt;a+b&lt;/code&gt;<a name="line.113"></a>
<FONT color="green">114</FONT>         * @throws ArithmeticException if the result can not be represented as an<a name="line.114"></a>
<FONT color="green">115</FONT>         *         long<a name="line.115"></a>
<FONT color="green">116</FONT>         * @since 1.2<a name="line.116"></a>
<FONT color="green">117</FONT>         */<a name="line.117"></a>
<FONT color="green">118</FONT>        private static long addAndCheck(long a, long b, String msg) {<a name="line.118"></a>
<FONT color="green">119</FONT>            long ret;<a name="line.119"></a>
<FONT color="green">120</FONT>            if (a &gt; b) {<a name="line.120"></a>
<FONT color="green">121</FONT>                // use symmetry to reduce boundary cases<a name="line.121"></a>
<FONT color="green">122</FONT>                ret = addAndCheck(b, a, msg);<a name="line.122"></a>
<FONT color="green">123</FONT>            } else {<a name="line.123"></a>
<FONT color="green">124</FONT>                // assert a &lt;= b<a name="line.124"></a>
<FONT color="green">125</FONT>                <a name="line.125"></a>
<FONT color="green">126</FONT>                if (a &lt; 0) {<a name="line.126"></a>
<FONT color="green">127</FONT>                    if (b &lt; 0) {<a name="line.127"></a>
<FONT color="green">128</FONT>                        // check for negative overflow<a name="line.128"></a>
<FONT color="green">129</FONT>                        if (Long.MIN_VALUE - b &lt;= a) {<a name="line.129"></a>
<FONT color="green">130</FONT>                            ret = a + b;<a name="line.130"></a>
<FONT color="green">131</FONT>                        } else {<a name="line.131"></a>
<FONT color="green">132</FONT>                            throw new ArithmeticException(msg);<a name="line.132"></a>
<FONT color="green">133</FONT>                        }<a name="line.133"></a>
<FONT color="green">134</FONT>                    } else {<a name="line.134"></a>
<FONT color="green">135</FONT>                        // opposite sign addition is always safe<a name="line.135"></a>
<FONT color="green">136</FONT>                        ret = a + b;<a name="line.136"></a>
<FONT color="green">137</FONT>                    }<a name="line.137"></a>
<FONT color="green">138</FONT>                } else {<a name="line.138"></a>
<FONT color="green">139</FONT>                    // assert a &gt;= 0<a name="line.139"></a>
<FONT color="green">140</FONT>                    // assert b &gt;= 0<a name="line.140"></a>
<FONT color="green">141</FONT>    <a name="line.141"></a>
<FONT color="green">142</FONT>                    // check for positive overflow<a name="line.142"></a>
<FONT color="green">143</FONT>                    if (a &lt;= Long.MAX_VALUE - b) {<a name="line.143"></a>
<FONT color="green">144</FONT>                        ret = a + b;<a name="line.144"></a>
<FONT color="green">145</FONT>                    } else {<a name="line.145"></a>
<FONT color="green">146</FONT>                        throw new ArithmeticException(msg);<a name="line.146"></a>
<FONT color="green">147</FONT>                    }<a name="line.147"></a>
<FONT color="green">148</FONT>                }<a name="line.148"></a>
<FONT color="green">149</FONT>            }<a name="line.149"></a>
<FONT color="green">150</FONT>            return ret;<a name="line.150"></a>
<FONT color="green">151</FONT>        }<a name="line.151"></a>
<FONT color="green">152</FONT>        <a name="line.152"></a>
<FONT color="green">153</FONT>        /**<a name="line.153"></a>
<FONT color="green">154</FONT>         * Returns an exact representation of the &lt;a<a name="line.154"></a>
<FONT color="green">155</FONT>         * href="http://mathworld.wolfram.com/BinomialCoefficient.html"&gt; Binomial<a name="line.155"></a>
<FONT color="green">156</FONT>         * Coefficient&lt;/a&gt;, "&lt;code&gt;n choose k&lt;/code&gt;", the number of<a name="line.156"></a>
<FONT color="green">157</FONT>         * &lt;code&gt;k&lt;/code&gt;-element subsets that can be selected from an<a name="line.157"></a>
<FONT color="green">158</FONT>         * &lt;code&gt;n&lt;/code&gt;-element set.<a name="line.158"></a>
<FONT color="green">159</FONT>         * &lt;p&gt;<a name="line.159"></a>
<FONT color="green">160</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.160"></a>
<FONT color="green">161</FONT>         * &lt;ul&gt;<a name="line.161"></a>
<FONT color="green">162</FONT>         * &lt;li&gt; &lt;code&gt;0 &lt;= k &lt;= n &lt;/code&gt; (otherwise<a name="line.162"></a>
<FONT color="green">163</FONT>         * &lt;code&gt;IllegalArgumentException&lt;/code&gt; is thrown)&lt;/li&gt;<a name="line.163"></a>
<FONT color="green">164</FONT>         * &lt;li&gt; The result is small enough to fit into a &lt;code&gt;long&lt;/code&gt;. The<a name="line.164"></a>
<FONT color="green">165</FONT>         * largest value of &lt;code&gt;n&lt;/code&gt; for which all coefficients are<a name="line.165"></a>
<FONT color="green">166</FONT>         * &lt;code&gt; &lt; Long.MAX_VALUE&lt;/code&gt; is 66. If the computed value exceeds<a name="line.166"></a>
<FONT color="green">167</FONT>         * &lt;code&gt;Long.MAX_VALUE&lt;/code&gt; an &lt;code&gt;ArithMeticException&lt;/code&gt; is<a name="line.167"></a>
<FONT color="green">168</FONT>         * thrown.&lt;/li&gt;<a name="line.168"></a>
<FONT color="green">169</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.169"></a>
<FONT color="green">170</FONT>         * <a name="line.170"></a>
<FONT color="green">171</FONT>         * @param n the size of the set<a name="line.171"></a>
<FONT color="green">172</FONT>         * @param k the size of the subsets to be counted<a name="line.172"></a>
<FONT color="green">173</FONT>         * @return &lt;code&gt;n choose k&lt;/code&gt;<a name="line.173"></a>
<FONT color="green">174</FONT>         * @throws IllegalArgumentException if preconditions are not met.<a name="line.174"></a>
<FONT color="green">175</FONT>         * @throws ArithmeticException if the result is too large to be represented<a name="line.175"></a>
<FONT color="green">176</FONT>         *         by a long integer.<a name="line.176"></a>
<FONT color="green">177</FONT>         */<a name="line.177"></a>
<FONT color="green">178</FONT>        public static long binomialCoefficient(final int n, final int k) {<a name="line.178"></a>
<FONT color="green">179</FONT>            checkBinomial(n, k);<a name="line.179"></a>
<FONT color="green">180</FONT>            if ((n == k) || (k == 0)) {<a name="line.180"></a>
<FONT color="green">181</FONT>                return 1;<a name="line.181"></a>
<FONT color="green">182</FONT>            }<a name="line.182"></a>
<FONT color="green">183</FONT>            if ((k == 1) || (k == n - 1)) {<a name="line.183"></a>
<FONT color="green">184</FONT>                return n;<a name="line.184"></a>
<FONT color="green">185</FONT>            }<a name="line.185"></a>
<FONT color="green">186</FONT>            // Use symmetry for large k<a name="line.186"></a>
<FONT color="green">187</FONT>            if (k &gt; n / 2)<a name="line.187"></a>
<FONT color="green">188</FONT>                return binomialCoefficient(n, n - k);<a name="line.188"></a>
<FONT color="green">189</FONT>            <a name="line.189"></a>
<FONT color="green">190</FONT>            // We use the formula<a name="line.190"></a>
<FONT color="green">191</FONT>            // (n choose k) = n! / (n-k)! / k!<a name="line.191"></a>
<FONT color="green">192</FONT>            // (n choose k) == ((n-k+1)*...*n) / (1*...*k)<a name="line.192"></a>
<FONT color="green">193</FONT>            // which could be written<a name="line.193"></a>
<FONT color="green">194</FONT>            // (n choose k) == (n-1 choose k-1) * n / k<a name="line.194"></a>
<FONT color="green">195</FONT>            long result = 1;<a name="line.195"></a>
<FONT color="green">196</FONT>            if (n &lt;= 61) {<a name="line.196"></a>
<FONT color="green">197</FONT>                // For n &lt;= 61, the naive implementation cannot overflow.<a name="line.197"></a>
<FONT color="green">198</FONT>                for (int j = 1, i = n - k + 1; j &lt;= k; i++, j++) {<a name="line.198"></a>
<FONT color="green">199</FONT>                    result = result * i / j;<a name="line.199"></a>
<FONT color="green">200</FONT>                }<a name="line.200"></a>
<FONT color="green">201</FONT>            } else if (n &lt;= 66) {<a name="line.201"></a>
<FONT color="green">202</FONT>                // For n &gt; 61 but n &lt;= 66, the result cannot overflow,<a name="line.202"></a>
<FONT color="green">203</FONT>                // but we must take care not to overflow intermediate values.<a name="line.203"></a>
<FONT color="green">204</FONT>                for (int j = 1, i = n - k + 1; j &lt;= k; i++, j++) {<a name="line.204"></a>
<FONT color="green">205</FONT>                    // We know that (result * i) is divisible by j,<a name="line.205"></a>
<FONT color="green">206</FONT>                    // but (result * i) may overflow, so we split j:<a name="line.206"></a>
<FONT color="green">207</FONT>                    // Filter out the gcd, d, so j/d and i/d are integer.<a name="line.207"></a>
<FONT color="green">208</FONT>                    // result is divisible by (j/d) because (j/d)<a name="line.208"></a>
<FONT color="green">209</FONT>                    // is relative prime to (i/d) and is a divisor of<a name="line.209"></a>
<FONT color="green">210</FONT>                    // result * (i/d).<a name="line.210"></a>
<FONT color="green">211</FONT>                    long d = gcd(i, j);<a name="line.211"></a>
<FONT color="green">212</FONT>                    result = (result / (j / d)) * (i / d);<a name="line.212"></a>
<FONT color="green">213</FONT>                }<a name="line.213"></a>
<FONT color="green">214</FONT>            } else {<a name="line.214"></a>
<FONT color="green">215</FONT>                // For n &gt; 66, a result overflow might occur, so we check<a name="line.215"></a>
<FONT color="green">216</FONT>                // the multiplication, taking care to not overflow<a name="line.216"></a>
<FONT color="green">217</FONT>                // unnecessary.<a name="line.217"></a>
<FONT color="green">218</FONT>                for (int j = 1, i = n - k + 1; j &lt;= k; i++, j++) {<a name="line.218"></a>
<FONT color="green">219</FONT>                    long d = gcd(i, j);<a name="line.219"></a>
<FONT color="green">220</FONT>                    result = mulAndCheck((result / (j / d)), (i / d));<a name="line.220"></a>
<FONT color="green">221</FONT>                }<a name="line.221"></a>
<FONT color="green">222</FONT>            }<a name="line.222"></a>
<FONT color="green">223</FONT>            return result;<a name="line.223"></a>
<FONT color="green">224</FONT>        }<a name="line.224"></a>
<FONT color="green">225</FONT>    <a name="line.225"></a>
<FONT color="green">226</FONT>        /**<a name="line.226"></a>
<FONT color="green">227</FONT>         * Returns a &lt;code&gt;double&lt;/code&gt; representation of the &lt;a<a name="line.227"></a>
<FONT color="green">228</FONT>         * href="http://mathworld.wolfram.com/BinomialCoefficient.html"&gt; Binomial<a name="line.228"></a>
<FONT color="green">229</FONT>         * Coefficient&lt;/a&gt;, "&lt;code&gt;n choose k&lt;/code&gt;", the number of<a name="line.229"></a>
<FONT color="green">230</FONT>         * &lt;code&gt;k&lt;/code&gt;-element subsets that can be selected from an<a name="line.230"></a>
<FONT color="green">231</FONT>         * &lt;code&gt;n&lt;/code&gt;-element set.<a name="line.231"></a>
<FONT color="green">232</FONT>         * &lt;p&gt;<a name="line.232"></a>
<FONT color="green">233</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.233"></a>
<FONT color="green">234</FONT>         * &lt;ul&gt;<a name="line.234"></a>
<FONT color="green">235</FONT>         * &lt;li&gt; &lt;code&gt;0 &lt;= k &lt;= n &lt;/code&gt; (otherwise<a name="line.235"></a>
<FONT color="green">236</FONT>         * &lt;code&gt;IllegalArgumentException&lt;/code&gt; is thrown)&lt;/li&gt;<a name="line.236"></a>
<FONT color="green">237</FONT>         * &lt;li&gt; The result is small enough to fit into a &lt;code&gt;double&lt;/code&gt;. The<a name="line.237"></a>
<FONT color="green">238</FONT>         * largest value of &lt;code&gt;n&lt;/code&gt; for which all coefficients are &lt;<a name="line.238"></a>
<FONT color="green">239</FONT>         * Double.MAX_VALUE is 1029. If the computed value exceeds Double.MAX_VALUE,<a name="line.239"></a>
<FONT color="green">240</FONT>         * Double.POSITIVE_INFINITY is returned&lt;/li&gt;<a name="line.240"></a>
<FONT color="green">241</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.241"></a>
<FONT color="green">242</FONT>         * <a name="line.242"></a>
<FONT color="green">243</FONT>         * @param n the size of the set<a name="line.243"></a>
<FONT color="green">244</FONT>         * @param k the size of the subsets to be counted<a name="line.244"></a>
<FONT color="green">245</FONT>         * @return &lt;code&gt;n choose k&lt;/code&gt;<a name="line.245"></a>
<FONT color="green">246</FONT>         * @throws IllegalArgumentException if preconditions are not met.<a name="line.246"></a>
<FONT color="green">247</FONT>         */<a name="line.247"></a>
<FONT color="green">248</FONT>        public static double binomialCoefficientDouble(final int n, final int k) {<a name="line.248"></a>
<FONT color="green">249</FONT>            checkBinomial(n, k);<a name="line.249"></a>
<FONT color="green">250</FONT>            if ((n == k) || (k == 0)) {<a name="line.250"></a>
<FONT color="green">251</FONT>                return 1d;<a name="line.251"></a>
<FONT color="green">252</FONT>            }<a name="line.252"></a>
<FONT color="green">253</FONT>            if ((k == 1) || (k == n - 1)) {<a name="line.253"></a>
<FONT color="green">254</FONT>                return n;<a name="line.254"></a>
<FONT color="green">255</FONT>            }<a name="line.255"></a>
<FONT color="green">256</FONT>            if (k &gt; n/2) {<a name="line.256"></a>
<FONT color="green">257</FONT>                return binomialCoefficientDouble(n, n - k);<a name="line.257"></a>
<FONT color="green">258</FONT>            }<a name="line.258"></a>
<FONT color="green">259</FONT>            if (n &lt; 67) {<a name="line.259"></a>
<FONT color="green">260</FONT>                return binomialCoefficient(n,k);<a name="line.260"></a>
<FONT color="green">261</FONT>            }<a name="line.261"></a>
<FONT color="green">262</FONT>            <a name="line.262"></a>
<FONT color="green">263</FONT>            double result = 1d;<a name="line.263"></a>
<FONT color="green">264</FONT>            for (int i = 1; i &lt;= k; i++) {<a name="line.264"></a>
<FONT color="green">265</FONT>                 result *= (double)(n - k + i) / (double)i;<a name="line.265"></a>
<FONT color="green">266</FONT>            }<a name="line.266"></a>
<FONT color="green">267</FONT>      <a name="line.267"></a>
<FONT color="green">268</FONT>            return Math.floor(result + 0.5);<a name="line.268"></a>
<FONT color="green">269</FONT>        }<a name="line.269"></a>
<FONT color="green">270</FONT>        <a name="line.270"></a>
<FONT color="green">271</FONT>        /**<a name="line.271"></a>
<FONT color="green">272</FONT>         * Returns the natural &lt;code&gt;log&lt;/code&gt; of the &lt;a<a name="line.272"></a>
<FONT color="green">273</FONT>         * href="http://mathworld.wolfram.com/BinomialCoefficient.html"&gt; Binomial<a name="line.273"></a>
<FONT color="green">274</FONT>         * Coefficient&lt;/a&gt;, "&lt;code&gt;n choose k&lt;/code&gt;", the number of<a name="line.274"></a>
<FONT color="green">275</FONT>         * &lt;code&gt;k&lt;/code&gt;-element subsets that can be selected from an<a name="line.275"></a>
<FONT color="green">276</FONT>         * &lt;code&gt;n&lt;/code&gt;-element set.<a name="line.276"></a>
<FONT color="green">277</FONT>         * &lt;p&gt;<a name="line.277"></a>
<FONT color="green">278</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.278"></a>
<FONT color="green">279</FONT>         * &lt;ul&gt;<a name="line.279"></a>
<FONT color="green">280</FONT>         * &lt;li&gt; &lt;code&gt;0 &lt;= k &lt;= n &lt;/code&gt; (otherwise<a name="line.280"></a>
<FONT color="green">281</FONT>         * &lt;code&gt;IllegalArgumentException&lt;/code&gt; is thrown)&lt;/li&gt;<a name="line.281"></a>
<FONT color="green">282</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.282"></a>
<FONT color="green">283</FONT>         * <a name="line.283"></a>
<FONT color="green">284</FONT>         * @param n the size of the set<a name="line.284"></a>
<FONT color="green">285</FONT>         * @param k the size of the subsets to be counted<a name="line.285"></a>
<FONT color="green">286</FONT>         * @return &lt;code&gt;n choose k&lt;/code&gt;<a name="line.286"></a>
<FONT color="green">287</FONT>         * @throws IllegalArgumentException if preconditions are not met.<a name="line.287"></a>
<FONT color="green">288</FONT>         */<a name="line.288"></a>
<FONT color="green">289</FONT>        public static double binomialCoefficientLog(final int n, final int k) {<a name="line.289"></a>
<FONT color="green">290</FONT>            checkBinomial(n, k);<a name="line.290"></a>
<FONT color="green">291</FONT>            if ((n == k) || (k == 0)) {<a name="line.291"></a>
<FONT color="green">292</FONT>                return 0;<a name="line.292"></a>
<FONT color="green">293</FONT>            }<a name="line.293"></a>
<FONT color="green">294</FONT>            if ((k == 1) || (k == n - 1)) {<a name="line.294"></a>
<FONT color="green">295</FONT>                return Math.log(n);<a name="line.295"></a>
<FONT color="green">296</FONT>            }<a name="line.296"></a>
<FONT color="green">297</FONT>            <a name="line.297"></a>
<FONT color="green">298</FONT>            /*<a name="line.298"></a>
<FONT color="green">299</FONT>             * For values small enough to do exact integer computation,<a name="line.299"></a>
<FONT color="green">300</FONT>             * return the log of the exact value <a name="line.300"></a>
<FONT color="green">301</FONT>             */<a name="line.301"></a>
<FONT color="green">302</FONT>            if (n &lt; 67) {  <a name="line.302"></a>
<FONT color="green">303</FONT>                return Math.log(binomialCoefficient(n,k));<a name="line.303"></a>
<FONT color="green">304</FONT>            }<a name="line.304"></a>
<FONT color="green">305</FONT>            <a name="line.305"></a>
<FONT color="green">306</FONT>            /*<a name="line.306"></a>
<FONT color="green">307</FONT>             * Return the log of binomialCoefficientDouble for values that will not<a name="line.307"></a>
<FONT color="green">308</FONT>             * overflow binomialCoefficientDouble<a name="line.308"></a>
<FONT color="green">309</FONT>             */<a name="line.309"></a>
<FONT color="green">310</FONT>            if (n &lt; 1030) { <a name="line.310"></a>
<FONT color="green">311</FONT>                return Math.log(binomialCoefficientDouble(n, k));<a name="line.311"></a>
<FONT color="green">312</FONT>            } <a name="line.312"></a>
<FONT color="green">313</FONT>    <a name="line.313"></a>
<FONT color="green">314</FONT>            if (k &gt; n / 2) {<a name="line.314"></a>
<FONT color="green">315</FONT>                return binomialCoefficientLog(n, n - k);<a name="line.315"></a>
<FONT color="green">316</FONT>            }<a name="line.316"></a>
<FONT color="green">317</FONT>    <a name="line.317"></a>
<FONT color="green">318</FONT>            /*<a name="line.318"></a>
<FONT color="green">319</FONT>             * Sum logs for values that could overflow<a name="line.319"></a>
<FONT color="green">320</FONT>             */<a name="line.320"></a>
<FONT color="green">321</FONT>            double logSum = 0;<a name="line.321"></a>
<FONT color="green">322</FONT>    <a name="line.322"></a>
<FONT color="green">323</FONT>            // n!/(n-k)!<a name="line.323"></a>
<FONT color="green">324</FONT>            for (int i = n - k + 1; i &lt;= n; i++) {<a name="line.324"></a>
<FONT color="green">325</FONT>                logSum += Math.log(i);<a name="line.325"></a>
<FONT color="green">326</FONT>            }<a name="line.326"></a>
<FONT color="green">327</FONT>    <a name="line.327"></a>
<FONT color="green">328</FONT>            // divide by k!<a name="line.328"></a>
<FONT color="green">329</FONT>            for (int i = 2; i &lt;= k; i++) {<a name="line.329"></a>
<FONT color="green">330</FONT>                logSum -= Math.log(i);<a name="line.330"></a>
<FONT color="green">331</FONT>            }<a name="line.331"></a>
<FONT color="green">332</FONT>    <a name="line.332"></a>
<FONT color="green">333</FONT>            return logSum;      <a name="line.333"></a>
<FONT color="green">334</FONT>        }<a name="line.334"></a>
<FONT color="green">335</FONT>    <a name="line.335"></a>
<FONT color="green">336</FONT>        /**<a name="line.336"></a>
<FONT color="green">337</FONT>         * Check binomial preconditions.<a name="line.337"></a>
<FONT color="green">338</FONT>         * @param n the size of the set<a name="line.338"></a>
<FONT color="green">339</FONT>         * @param k the size of the subsets to be counted<a name="line.339"></a>
<FONT color="green">340</FONT>         * @exception IllegalArgumentException if preconditions are not met.<a name="line.340"></a>
<FONT color="green">341</FONT>         */<a name="line.341"></a>
<FONT color="green">342</FONT>        private static void checkBinomial(final int n, final int k)<a name="line.342"></a>
<FONT color="green">343</FONT>            throws IllegalArgumentException {<a name="line.343"></a>
<FONT color="green">344</FONT>            if (n &lt; k) {<a name="line.344"></a>
<FONT color="green">345</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.345"></a>
<FONT color="green">346</FONT>                    "must have n &gt;= k for binomial coefficient (n,k), got n = {0}, k = {1}",<a name="line.346"></a>
<FONT color="green">347</FONT>                    n, k);<a name="line.347"></a>
<FONT color="green">348</FONT>            }<a name="line.348"></a>
<FONT color="green">349</FONT>            if (n &lt; 0) {<a name="line.349"></a>
<FONT color="green">350</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.350"></a>
<FONT color="green">351</FONT>                      "must have n &gt;= 0 for binomial coefficient (n,k), got n = {0}",<a name="line.351"></a>
<FONT color="green">352</FONT>                      n);<a name="line.352"></a>
<FONT color="green">353</FONT>            }<a name="line.353"></a>
<FONT color="green">354</FONT>        }<a name="line.354"></a>
<FONT color="green">355</FONT>        <a name="line.355"></a>
<FONT color="green">356</FONT>        /**<a name="line.356"></a>
<FONT color="green">357</FONT>         * Compares two numbers given some amount of allowed error.<a name="line.357"></a>
<FONT color="green">358</FONT>         * <a name="line.358"></a>
<FONT color="green">359</FONT>         * @param x the first number<a name="line.359"></a>
<FONT color="green">360</FONT>         * @param y the second number<a name="line.360"></a>
<FONT color="green">361</FONT>         * @param eps the amount of error to allow when checking for equality<a name="line.361"></a>
<FONT color="green">362</FONT>         * @return &lt;ul&gt;&lt;li&gt;0 if  {@link #equals(double, double, double) equals(x, y, eps)}&lt;/li&gt;<a name="line.362"></a>
<FONT color="green">363</FONT>         *       &lt;li&gt;&amp;lt; 0 if !{@link #equals(double, double, double) equals(x, y, eps)} &amp;amp;&amp;amp; x &amp;lt; y&lt;/li&gt;<a name="line.363"></a>
<FONT color="green">364</FONT>         *       &lt;li&gt;&gt; 0 if !{@link #equals(double, double, double) equals(x, y, eps)} &amp;amp;&amp;amp; x &gt; y&lt;/li&gt;&lt;/ul&gt;<a name="line.364"></a>
<FONT color="green">365</FONT>         */<a name="line.365"></a>
<FONT color="green">366</FONT>        public static int compareTo(double x, double y, double eps) {<a name="line.366"></a>
<FONT color="green">367</FONT>            if (equals(x, y, eps)) {<a name="line.367"></a>
<FONT color="green">368</FONT>                return 0;<a name="line.368"></a>
<FONT color="green">369</FONT>            } else if (x &lt; y) {<a name="line.369"></a>
<FONT color="green">370</FONT>              return -1;<a name="line.370"></a>
<FONT color="green">371</FONT>            }<a name="line.371"></a>
<FONT color="green">372</FONT>            return 1;<a name="line.372"></a>
<FONT color="green">373</FONT>        }<a name="line.373"></a>
<FONT color="green">374</FONT>        <a name="line.374"></a>
<FONT color="green">375</FONT>        /**<a name="line.375"></a>
<FONT color="green">376</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/HyperbolicCosine.html"&gt;<a name="line.376"></a>
<FONT color="green">377</FONT>         * hyperbolic cosine&lt;/a&gt; of x.<a name="line.377"></a>
<FONT color="green">378</FONT>         * <a name="line.378"></a>
<FONT color="green">379</FONT>         * @param x double value for which to find the hyperbolic cosine<a name="line.379"></a>
<FONT color="green">380</FONT>         * @return hyperbolic cosine of x<a name="line.380"></a>
<FONT color="green">381</FONT>         */<a name="line.381"></a>
<FONT color="green">382</FONT>        public static double cosh(double x) {<a name="line.382"></a>
<FONT color="green">383</FONT>            return (Math.exp(x) + Math.exp(-x)) / 2.0;<a name="line.383"></a>
<FONT color="green">384</FONT>        }<a name="line.384"></a>
<FONT color="green">385</FONT>        <a name="line.385"></a>
<FONT color="green">386</FONT>        /**<a name="line.386"></a>
<FONT color="green">387</FONT>         * Returns true iff both arguments are NaN or neither is NaN and they are<a name="line.387"></a>
<FONT color="green">388</FONT>         * equal<a name="line.388"></a>
<FONT color="green">389</FONT>         * <a name="line.389"></a>
<FONT color="green">390</FONT>         * @param x first value<a name="line.390"></a>
<FONT color="green">391</FONT>         * @param y second value<a name="line.391"></a>
<FONT color="green">392</FONT>         * @return true if the values are equal or both are NaN<a name="line.392"></a>
<FONT color="green">393</FONT>         */<a name="line.393"></a>
<FONT color="green">394</FONT>        public static boolean equals(double x, double y) {<a name="line.394"></a>
<FONT color="green">395</FONT>            return ((Double.isNaN(x) &amp;&amp; Double.isNaN(y)) || x == y);<a name="line.395"></a>
<FONT color="green">396</FONT>        }<a name="line.396"></a>
<FONT color="green">397</FONT>    <a name="line.397"></a>
<FONT color="green">398</FONT>        /**<a name="line.398"></a>
<FONT color="green">399</FONT>         * Returns true iff both arguments are equal or within the range of allowed<a name="line.399"></a>
<FONT color="green">400</FONT>         * error (inclusive).<a name="line.400"></a>
<FONT color="green">401</FONT>         * &lt;p&gt;<a name="line.401"></a>
<FONT color="green">402</FONT>         * Two NaNs are considered equals, as are two infinities with same sign.<a name="line.402"></a>
<FONT color="green">403</FONT>         * &lt;/p&gt;<a name="line.403"></a>
<FONT color="green">404</FONT>         * <a name="line.404"></a>
<FONT color="green">405</FONT>         * @param x first value<a name="line.405"></a>
<FONT color="green">406</FONT>         * @param y second value<a name="line.406"></a>
<FONT color="green">407</FONT>         * @param eps the amount of absolute error to allow<a name="line.407"></a>
<FONT color="green">408</FONT>         * @return true if the values are equal or within range of each other<a name="line.408"></a>
<FONT color="green">409</FONT>         */<a name="line.409"></a>
<FONT color="green">410</FONT>        public static boolean equals(double x, double y, double eps) {<a name="line.410"></a>
<FONT color="green">411</FONT>          return equals(x, y) || (Math.abs(y - x) &lt;= eps);<a name="line.411"></a>
<FONT color="green">412</FONT>        }<a name="line.412"></a>
<FONT color="green">413</FONT>        <a name="line.413"></a>
<FONT color="green">414</FONT>        /**<a name="line.414"></a>
<FONT color="green">415</FONT>         * Returns true iff both arguments are equal or within the range of allowed<a name="line.415"></a>
<FONT color="green">416</FONT>         * error (inclusive).<a name="line.416"></a>
<FONT color="green">417</FONT>         * Adapted from &lt;a<a name="line.417"></a>
<FONT color="green">418</FONT>         * href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm"&gt;<a name="line.418"></a>
<FONT color="green">419</FONT>         * Bruce Dawson&lt;/a&gt;<a name="line.419"></a>
<FONT color="green">420</FONT>         *<a name="line.420"></a>
<FONT color="green">421</FONT>         * @param x first value<a name="line.421"></a>
<FONT color="green">422</FONT>         * @param y second value<a name="line.422"></a>
<FONT color="green">423</FONT>         * @param maxUlps {@code (maxUlps - 1)} is the number of floating point<a name="line.423"></a>
<FONT color="green">424</FONT>         * values between {@code x} and {@code y}.<a name="line.424"></a>
<FONT color="green">425</FONT>         * @return {@code true} if there are less than {@code maxUlps} floating<a name="line.425"></a>
<FONT color="green">426</FONT>         * point values between {@code x} and {@code y}<a name="line.426"></a>
<FONT color="green">427</FONT>         */<a name="line.427"></a>
<FONT color="green">428</FONT>        public static boolean equals(double x, double y, int maxUlps) {<a name="line.428"></a>
<FONT color="green">429</FONT>            // Check that "maxUlps" is non-negative and small enough so that the<a name="line.429"></a>
<FONT color="green">430</FONT>            // default NAN won't compare as equal to anything.<a name="line.430"></a>
<FONT color="green">431</FONT>            assert maxUlps &gt; 0 &amp;&amp; maxUlps &lt; NAN_GAP;<a name="line.431"></a>
<FONT color="green">432</FONT>    <a name="line.432"></a>
<FONT color="green">433</FONT>            long xInt = Double.doubleToLongBits(x);<a name="line.433"></a>
<FONT color="green">434</FONT>            long yInt = Double.doubleToLongBits(y);<a name="line.434"></a>
<FONT color="green">435</FONT>    <a name="line.435"></a>
<FONT color="green">436</FONT>            // Make lexicographically ordered as a two's-complement integer.<a name="line.436"></a>
<FONT color="green">437</FONT>            if (xInt &lt; 0) {<a name="line.437"></a>
<FONT color="green">438</FONT>                xInt = SGN_MASK - xInt;<a name="line.438"></a>
<FONT color="green">439</FONT>            }<a name="line.439"></a>
<FONT color="green">440</FONT>            if (yInt &lt; 0) {<a name="line.440"></a>
<FONT color="green">441</FONT>                yInt = SGN_MASK - yInt;<a name="line.441"></a>
<FONT color="green">442</FONT>            }<a name="line.442"></a>
<FONT color="green">443</FONT>    <a name="line.443"></a>
<FONT color="green">444</FONT>            return Math.abs(xInt - yInt) &lt;= maxUlps;<a name="line.444"></a>
<FONT color="green">445</FONT>        }<a name="line.445"></a>
<FONT color="green">446</FONT>    <a name="line.446"></a>
<FONT color="green">447</FONT>        /**<a name="line.447"></a>
<FONT color="green">448</FONT>         * Returns true iff both arguments are null or have same dimensions<a name="line.448"></a>
<FONT color="green">449</FONT>         * and all their elements are {@link #equals(double,double) equals}<a name="line.449"></a>
<FONT color="green">450</FONT>         * <a name="line.450"></a>
<FONT color="green">451</FONT>         * @param x first array<a name="line.451"></a>
<FONT color="green">452</FONT>         * @param y second array<a name="line.452"></a>
<FONT color="green">453</FONT>         * @return true if the values are both null or have same dimension<a name="line.453"></a>
<FONT color="green">454</FONT>         * and equal elements<a name="line.454"></a>
<FONT color="green">455</FONT>         * @since 1.2<a name="line.455"></a>
<FONT color="green">456</FONT>         */<a name="line.456"></a>
<FONT color="green">457</FONT>        public static boolean equals(double[] x, double[] y) {<a name="line.457"></a>
<FONT color="green">458</FONT>            if ((x == null) || (y == null)) {<a name="line.458"></a>
<FONT color="green">459</FONT>                return !((x == null) ^ (y == null));<a name="line.459"></a>
<FONT color="green">460</FONT>            }<a name="line.460"></a>
<FONT color="green">461</FONT>            if (x.length != y.length) {<a name="line.461"></a>
<FONT color="green">462</FONT>                return false;<a name="line.462"></a>
<FONT color="green">463</FONT>            }<a name="line.463"></a>
<FONT color="green">464</FONT>            for (int i = 0; i &lt; x.length; ++i) {<a name="line.464"></a>
<FONT color="green">465</FONT>                if (!equals(x[i], y[i])) {<a name="line.465"></a>
<FONT color="green">466</FONT>                    return false;<a name="line.466"></a>
<FONT color="green">467</FONT>                }<a name="line.467"></a>
<FONT color="green">468</FONT>            }<a name="line.468"></a>
<FONT color="green">469</FONT>            return true;<a name="line.469"></a>
<FONT color="green">470</FONT>        }<a name="line.470"></a>
<FONT color="green">471</FONT>        <a name="line.471"></a>
<FONT color="green">472</FONT>        /** All long-representable factorials */<a name="line.472"></a>
<FONT color="green">473</FONT>        private static final long[] factorials = new long[] <a name="line.473"></a>
<FONT color="green">474</FONT>           {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800,<a name="line.474"></a>
<FONT color="green">475</FONT>            479001600, 6227020800l, 87178291200l, 1307674368000l, 20922789888000l,<a name="line.475"></a>
<FONT color="green">476</FONT>            355687428096000l, 6402373705728000l, 121645100408832000l,<a name="line.476"></a>
<FONT color="green">477</FONT>            2432902008176640000l};<a name="line.477"></a>
<FONT color="green">478</FONT>    <a name="line.478"></a>
<FONT color="green">479</FONT>        /**<a name="line.479"></a>
<FONT color="green">480</FONT>         * Returns n!. Shorthand for &lt;code&gt;n&lt;/code&gt; &lt;a<a name="line.480"></a>
<FONT color="green">481</FONT>         * href="http://mathworld.wolfram.com/Factorial.html"&gt; Factorial&lt;/a&gt;, the<a name="line.481"></a>
<FONT color="green">482</FONT>         * product of the numbers &lt;code&gt;1,...,n&lt;/code&gt;.<a name="line.482"></a>
<FONT color="green">483</FONT>         * &lt;p&gt;<a name="line.483"></a>
<FONT color="green">484</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.484"></a>
<FONT color="green">485</FONT>         * &lt;ul&gt;<a name="line.485"></a>
<FONT color="green">486</FONT>         * &lt;li&gt; &lt;code&gt;n &gt;= 0&lt;/code&gt; (otherwise<a name="line.486"></a>
<FONT color="green">487</FONT>         * &lt;code&gt;IllegalArgumentException&lt;/code&gt; is thrown)&lt;/li&gt;<a name="line.487"></a>
<FONT color="green">488</FONT>         * &lt;li&gt; The result is small enough to fit into a &lt;code&gt;long&lt;/code&gt;. The<a name="line.488"></a>
<FONT color="green">489</FONT>         * largest value of &lt;code&gt;n&lt;/code&gt; for which &lt;code&gt;n!&lt;/code&gt; &lt;<a name="line.489"></a>
<FONT color="green">490</FONT>         * Long.MAX_VALUE&lt;/code&gt; is 20. If the computed value exceeds &lt;code&gt;Long.MAX_VALUE&lt;/code&gt;<a name="line.490"></a>
<FONT color="green">491</FONT>         * an &lt;code&gt;ArithMeticException &lt;/code&gt; is thrown.&lt;/li&gt;<a name="line.491"></a>
<FONT color="green">492</FONT>         * &lt;/ul&gt;<a name="line.492"></a>
<FONT color="green">493</FONT>         * &lt;/p&gt;<a name="line.493"></a>
<FONT color="green">494</FONT>         * <a name="line.494"></a>
<FONT color="green">495</FONT>         * @param n argument<a name="line.495"></a>
<FONT color="green">496</FONT>         * @return &lt;code&gt;n!&lt;/code&gt;<a name="line.496"></a>
<FONT color="green">497</FONT>         * @throws ArithmeticException if the result is too large to be represented<a name="line.497"></a>
<FONT color="green">498</FONT>         *         by a long integer.<a name="line.498"></a>
<FONT color="green">499</FONT>         * @throws IllegalArgumentException if n &lt; 0<a name="line.499"></a>
<FONT color="green">500</FONT>         */<a name="line.500"></a>
<FONT color="green">501</FONT>        public static long factorial(final int n) {<a name="line.501"></a>
<FONT color="green">502</FONT>            if (n &lt; 0) {<a name="line.502"></a>
<FONT color="green">503</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.503"></a>
<FONT color="green">504</FONT>                      "must have n &gt;= 0 for n!, got n = {0}",<a name="line.504"></a>
<FONT color="green">505</FONT>                      n);<a name="line.505"></a>
<FONT color="green">506</FONT>            }<a name="line.506"></a>
<FONT color="green">507</FONT>            if (n &gt; 20) {<a name="line.507"></a>
<FONT color="green">508</FONT>                throw new ArithmeticException(<a name="line.508"></a>
<FONT color="green">509</FONT>                        "factorial value is too large to fit in a long");<a name="line.509"></a>
<FONT color="green">510</FONT>            }<a name="line.510"></a>
<FONT color="green">511</FONT>            return factorials[n];<a name="line.511"></a>
<FONT color="green">512</FONT>        }<a name="line.512"></a>
<FONT color="green">513</FONT>    <a name="line.513"></a>
<FONT color="green">514</FONT>        /**<a name="line.514"></a>
<FONT color="green">515</FONT>         * Returns n!. Shorthand for &lt;code&gt;n&lt;/code&gt; &lt;a<a name="line.515"></a>
<FONT color="green">516</FONT>         * href="http://mathworld.wolfram.com/Factorial.html"&gt; Factorial&lt;/a&gt;, the<a name="line.516"></a>
<FONT color="green">517</FONT>         * product of the numbers &lt;code&gt;1,...,n&lt;/code&gt; as a &lt;code&gt;double&lt;/code&gt;.<a name="line.517"></a>
<FONT color="green">518</FONT>         * &lt;p&gt;<a name="line.518"></a>
<FONT color="green">519</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.519"></a>
<FONT color="green">520</FONT>         * &lt;ul&gt;<a name="line.520"></a>
<FONT color="green">521</FONT>         * &lt;li&gt; &lt;code&gt;n &gt;= 0&lt;/code&gt; (otherwise<a name="line.521"></a>
<FONT color="green">522</FONT>         * &lt;code&gt;IllegalArgumentException&lt;/code&gt; is thrown)&lt;/li&gt;<a name="line.522"></a>
<FONT color="green">523</FONT>         * &lt;li&gt; The result is small enough to fit into a &lt;code&gt;double&lt;/code&gt;. The<a name="line.523"></a>
<FONT color="green">524</FONT>         * largest value of &lt;code&gt;n&lt;/code&gt; for which &lt;code&gt;n!&lt;/code&gt; &lt;<a name="line.524"></a>
<FONT color="green">525</FONT>         * Double.MAX_VALUE&lt;/code&gt; is 170. If the computed value exceeds<a name="line.525"></a>
<FONT color="green">526</FONT>         * Double.MAX_VALUE, Double.POSITIVE_INFINITY is returned&lt;/li&gt;<a name="line.526"></a>
<FONT color="green">527</FONT>         * &lt;/ul&gt;<a name="line.527"></a>
<FONT color="green">528</FONT>         * &lt;/p&gt;<a name="line.528"></a>
<FONT color="green">529</FONT>         * <a name="line.529"></a>
<FONT color="green">530</FONT>         * @param n argument<a name="line.530"></a>
<FONT color="green">531</FONT>         * @return &lt;code&gt;n!&lt;/code&gt;<a name="line.531"></a>
<FONT color="green">532</FONT>         * @throws IllegalArgumentException if n &lt; 0<a name="line.532"></a>
<FONT color="green">533</FONT>         */<a name="line.533"></a>
<FONT color="green">534</FONT>        public static double factorialDouble(final int n) {<a name="line.534"></a>
<FONT color="green">535</FONT>            if (n &lt; 0) {<a name="line.535"></a>
<FONT color="green">536</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.536"></a>
<FONT color="green">537</FONT>                      "must have n &gt;= 0 for n!, got n = {0}",<a name="line.537"></a>
<FONT color="green">538</FONT>                      n);<a name="line.538"></a>
<FONT color="green">539</FONT>            }<a name="line.539"></a>
<FONT color="green">540</FONT>            if (n &lt; 21) {<a name="line.540"></a>
<FONT color="green">541</FONT>                return factorial(n);<a name="line.541"></a>
<FONT color="green">542</FONT>            }<a name="line.542"></a>
<FONT color="green">543</FONT>            return Math.floor(Math.exp(factorialLog(n)) + 0.5);<a name="line.543"></a>
<FONT color="green">544</FONT>        }<a name="line.544"></a>
<FONT color="green">545</FONT>    <a name="line.545"></a>
<FONT color="green">546</FONT>        /**<a name="line.546"></a>
<FONT color="green">547</FONT>         * Returns the natural logarithm of n!.<a name="line.547"></a>
<FONT color="green">548</FONT>         * &lt;p&gt;<a name="line.548"></a>
<FONT color="green">549</FONT>         * &lt;Strong&gt;Preconditions&lt;/strong&gt;:<a name="line.549"></a>
<FONT color="green">550</FONT>         * &lt;ul&gt;<a name="line.550"></a>
<FONT color="green">551</FONT>         * &lt;li&gt; &lt;code&gt;n &gt;= 0&lt;/code&gt; (otherwise<a name="line.551"></a>
<FONT color="green">552</FONT>         * &lt;code&gt;IllegalArgumentException&lt;/code&gt; is thrown)&lt;/li&gt;<a name="line.552"></a>
<FONT color="green">553</FONT>         * &lt;/ul&gt;&lt;/p&gt;<a name="line.553"></a>
<FONT color="green">554</FONT>         * <a name="line.554"></a>
<FONT color="green">555</FONT>         * @param n argument<a name="line.555"></a>
<FONT color="green">556</FONT>         * @return &lt;code&gt;n!&lt;/code&gt;<a name="line.556"></a>
<FONT color="green">557</FONT>         * @throws IllegalArgumentException if preconditions are not met.<a name="line.557"></a>
<FONT color="green">558</FONT>         */<a name="line.558"></a>
<FONT color="green">559</FONT>        public static double factorialLog(final int n) {<a name="line.559"></a>
<FONT color="green">560</FONT>            if (n &lt; 0) {<a name="line.560"></a>
<FONT color="green">561</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.561"></a>
<FONT color="green">562</FONT>                      "must have n &gt;= 0 for n!, got n = {0}",<a name="line.562"></a>
<FONT color="green">563</FONT>                      n);<a name="line.563"></a>
<FONT color="green">564</FONT>            }<a name="line.564"></a>
<FONT color="green">565</FONT>            if (n &lt; 21) {<a name="line.565"></a>
<FONT color="green">566</FONT>                return Math.log(factorial(n));<a name="line.566"></a>
<FONT color="green">567</FONT>            }<a name="line.567"></a>
<FONT color="green">568</FONT>            double logSum = 0;<a name="line.568"></a>
<FONT color="green">569</FONT>            for (int i = 2; i &lt;= n; i++) {<a name="line.569"></a>
<FONT color="green">570</FONT>                logSum += Math.log(i);<a name="line.570"></a>
<FONT color="green">571</FONT>            }<a name="line.571"></a>
<FONT color="green">572</FONT>            return logSum;<a name="line.572"></a>
<FONT color="green">573</FONT>        }<a name="line.573"></a>
<FONT color="green">574</FONT>    <a name="line.574"></a>
<FONT color="green">575</FONT>        /**<a name="line.575"></a>
<FONT color="green">576</FONT>         * &lt;p&gt;<a name="line.576"></a>
<FONT color="green">577</FONT>         * Gets the greatest common divisor of the absolute value of two numbers,<a name="line.577"></a>
<FONT color="green">578</FONT>         * using the "binary gcd" method which avoids division and modulo<a name="line.578"></a>
<FONT color="green">579</FONT>         * operations. See Knuth 4.5.2 algorithm B. This algorithm is due to Josef<a name="line.579"></a>
<FONT color="green">580</FONT>         * Stein (1961).<a name="line.580"></a>
<FONT color="green">581</FONT>         * &lt;/p&gt;<a name="line.581"></a>
<FONT color="green">582</FONT>         * Special cases:<a name="line.582"></a>
<FONT color="green">583</FONT>         * &lt;ul&gt;<a name="line.583"></a>
<FONT color="green">584</FONT>         * &lt;li&gt;The invocations<a name="line.584"></a>
<FONT color="green">585</FONT>         * &lt;code&gt;gcd(Integer.MIN_VALUE, Integer.MIN_VALUE)&lt;/code&gt;,<a name="line.585"></a>
<FONT color="green">586</FONT>         * &lt;code&gt;gcd(Integer.MIN_VALUE, 0)&lt;/code&gt; and<a name="line.586"></a>
<FONT color="green">587</FONT>         * &lt;code&gt;gcd(0, Integer.MIN_VALUE)&lt;/code&gt; throw an<a name="line.587"></a>
<FONT color="green">588</FONT>         * &lt;code&gt;ArithmeticException&lt;/code&gt;, because the result would be 2^31, which<a name="line.588"></a>
<FONT color="green">589</FONT>         * is too large for an int value.&lt;/li&gt;<a name="line.589"></a>
<FONT color="green">590</FONT>         * &lt;li&gt;The result of &lt;code&gt;gcd(x, x)&lt;/code&gt;, &lt;code&gt;gcd(0, x)&lt;/code&gt; and<a name="line.590"></a>
<FONT color="green">591</FONT>         * &lt;code&gt;gcd(x, 0)&lt;/code&gt; is the absolute value of &lt;code&gt;x&lt;/code&gt;, except<a name="line.591"></a>
<FONT color="green">592</FONT>         * for the special cases above.<a name="line.592"></a>
<FONT color="green">593</FONT>         * &lt;li&gt;The invocation &lt;code&gt;gcd(0, 0)&lt;/code&gt; is the only one which returns<a name="line.593"></a>
<FONT color="green">594</FONT>         * &lt;code&gt;0&lt;/code&gt;.&lt;/li&gt;<a name="line.594"></a>
<FONT color="green">595</FONT>         * &lt;/ul&gt;<a name="line.595"></a>
<FONT color="green">596</FONT>         * <a name="line.596"></a>
<FONT color="green">597</FONT>         * @param p any number<a name="line.597"></a>
<FONT color="green">598</FONT>         * @param q any number<a name="line.598"></a>
<FONT color="green">599</FONT>         * @return the greatest common divisor, never negative<a name="line.599"></a>
<FONT color="green">600</FONT>         * @throws ArithmeticException<a name="line.600"></a>
<FONT color="green">601</FONT>         *             if the result cannot be represented as a nonnegative int<a name="line.601"></a>
<FONT color="green">602</FONT>         *             value<a name="line.602"></a>
<FONT color="green">603</FONT>         * @since 1.1<a name="line.603"></a>
<FONT color="green">604</FONT>         */<a name="line.604"></a>
<FONT color="green">605</FONT>        public static int gcd(final int p, final int q) {<a name="line.605"></a>
<FONT color="green">606</FONT>            int u = p;<a name="line.606"></a>
<FONT color="green">607</FONT>            int v = q;<a name="line.607"></a>
<FONT color="green">608</FONT>            if ((u == 0) || (v == 0)) {<a name="line.608"></a>
<FONT color="green">609</FONT>                if ((u == Integer.MIN_VALUE) || (v == Integer.MIN_VALUE)) {<a name="line.609"></a>
<FONT color="green">610</FONT>                    throw MathRuntimeException.createArithmeticException(<a name="line.610"></a>
<FONT color="green">611</FONT>                            "overflow: gcd({0}, {1}) is 2^31",<a name="line.611"></a>
<FONT color="green">612</FONT>                            p, q);<a name="line.612"></a>
<FONT color="green">613</FONT>                }<a name="line.613"></a>
<FONT color="green">614</FONT>                return (Math.abs(u) + Math.abs(v));<a name="line.614"></a>
<FONT color="green">615</FONT>            }<a name="line.615"></a>
<FONT color="green">616</FONT>            // keep u and v negative, as negative integers range down to<a name="line.616"></a>
<FONT color="green">617</FONT>            // -2^31, while positive numbers can only be as large as 2^31-1<a name="line.617"></a>
<FONT color="green">618</FONT>            // (i.e. we can't necessarily negate a negative number without<a name="line.618"></a>
<FONT color="green">619</FONT>            // overflow)<a name="line.619"></a>
<FONT color="green">620</FONT>            /* assert u!=0 &amp;&amp; v!=0; */<a name="line.620"></a>
<FONT color="green">621</FONT>            if (u &gt; 0) {<a name="line.621"></a>
<FONT color="green">622</FONT>                u = -u;<a name="line.622"></a>
<FONT color="green">623</FONT>            } // make u negative<a name="line.623"></a>
<FONT color="green">624</FONT>            if (v &gt; 0) {<a name="line.624"></a>
<FONT color="green">625</FONT>                v = -v;<a name="line.625"></a>
<FONT color="green">626</FONT>            } // make v negative<a name="line.626"></a>
<FONT color="green">627</FONT>            // B1. [Find power of 2]<a name="line.627"></a>
<FONT color="green">628</FONT>            int k = 0;<a name="line.628"></a>
<FONT color="green">629</FONT>            while ((u &amp; 1) == 0 &amp;&amp; (v &amp; 1) == 0 &amp;&amp; k &lt; 31) { // while u and v are<a name="line.629"></a>
<FONT color="green">630</FONT>                                                                // both even...<a name="line.630"></a>
<FONT color="green">631</FONT>                u /= 2;<a name="line.631"></a>
<FONT color="green">632</FONT>                v /= 2;<a name="line.632"></a>
<FONT color="green">633</FONT>                k++; // cast out twos.<a name="line.633"></a>
<FONT color="green">634</FONT>            }<a name="line.634"></a>
<FONT color="green">635</FONT>            if (k == 31) {<a name="line.635"></a>
<FONT color="green">636</FONT>                throw MathRuntimeException.createArithmeticException(<a name="line.636"></a>
<FONT color="green">637</FONT>                        "overflow: gcd({0}, {1}) is 2^31",<a name="line.637"></a>
<FONT color="green">638</FONT>                        p, q);<a name="line.638"></a>
<FONT color="green">639</FONT>            }<a name="line.639"></a>
<FONT color="green">640</FONT>            // B2. Initialize: u and v have been divided by 2^k and at least<a name="line.640"></a>
<FONT color="green">641</FONT>            // one is odd.<a name="line.641"></a>
<FONT color="green">642</FONT>            int t = ((u &amp; 1) == 1) ? v : -(u / 2)/* B3 */;<a name="line.642"></a>
<FONT color="green">643</FONT>            // t negative: u was odd, v may be even (t replaces v)<a name="line.643"></a>
<FONT color="green">644</FONT>            // t positive: u was even, v is odd (t replaces u)<a name="line.644"></a>
<FONT color="green">645</FONT>            do {<a name="line.645"></a>
<FONT color="green">646</FONT>                /* assert u&lt;0 &amp;&amp; v&lt;0; */<a name="line.646"></a>
<FONT color="green">647</FONT>                // B4/B3: cast out twos from t.<a name="line.647"></a>
<FONT color="green">648</FONT>                while ((t &amp; 1) == 0) { // while t is even..<a name="line.648"></a>
<FONT color="green">649</FONT>                    t /= 2; // cast out twos<a name="line.649"></a>
<FONT color="green">650</FONT>                }<a name="line.650"></a>
<FONT color="green">651</FONT>                // B5 [reset max(u,v)]<a name="line.651"></a>
<FONT color="green">652</FONT>                if (t &gt; 0) {<a name="line.652"></a>
<FONT color="green">653</FONT>                    u = -t;<a name="line.653"></a>
<FONT color="green">654</FONT>                } else {<a name="line.654"></a>
<FONT color="green">655</FONT>                    v = t;<a name="line.655"></a>
<FONT color="green">656</FONT>                }<a name="line.656"></a>
<FONT color="green">657</FONT>                // B6/B3. at this point both u and v should be odd.<a name="line.657"></a>
<FONT color="green">658</FONT>                t = (v - u) / 2;<a name="line.658"></a>
<FONT color="green">659</FONT>                // |u| larger: t positive (replace u)<a name="line.659"></a>
<FONT color="green">660</FONT>                // |v| larger: t negative (replace v)<a name="line.660"></a>
<FONT color="green">661</FONT>            } while (t != 0);<a name="line.661"></a>
<FONT color="green">662</FONT>            return -u * (1 &lt;&lt; k); // gcd is u*2^k<a name="line.662"></a>
<FONT color="green">663</FONT>        }<a name="line.663"></a>
<FONT color="green">664</FONT>    <a name="line.664"></a>
<FONT color="green">665</FONT>        /**<a name="line.665"></a>
<FONT color="green">666</FONT>         * Returns an integer hash code representing the given double value.<a name="line.666"></a>
<FONT color="green">667</FONT>         * <a name="line.667"></a>
<FONT color="green">668</FONT>         * @param value the value to be hashed<a name="line.668"></a>
<FONT color="green">669</FONT>         * @return the hash code<a name="line.669"></a>
<FONT color="green">670</FONT>         */<a name="line.670"></a>
<FONT color="green">671</FONT>        public static int hash(double value) {<a name="line.671"></a>
<FONT color="green">672</FONT>            return new Double(value).hashCode();<a name="line.672"></a>
<FONT color="green">673</FONT>        }<a name="line.673"></a>
<FONT color="green">674</FONT>    <a name="line.674"></a>
<FONT color="green">675</FONT>        /**<a name="line.675"></a>
<FONT color="green">676</FONT>         * Returns an integer hash code representing the given double array.<a name="line.676"></a>
<FONT color="green">677</FONT>         * <a name="line.677"></a>
<FONT color="green">678</FONT>         * @param value the value to be hashed (may be null)<a name="line.678"></a>
<FONT color="green">679</FONT>         * @return the hash code<a name="line.679"></a>
<FONT color="green">680</FONT>         * @since 1.2<a name="line.680"></a>
<FONT color="green">681</FONT>         */<a name="line.681"></a>
<FONT color="green">682</FONT>        public static int hash(double[] value) {<a name="line.682"></a>
<FONT color="green">683</FONT>            return Arrays.hashCode(value);<a name="line.683"></a>
<FONT color="green">684</FONT>        }<a name="line.684"></a>
<FONT color="green">685</FONT>    <a name="line.685"></a>
<FONT color="green">686</FONT>        /**<a name="line.686"></a>
<FONT color="green">687</FONT>         * For a byte value x, this method returns (byte)(+1) if x &gt;= 0 and<a name="line.687"></a>
<FONT color="green">688</FONT>         * (byte)(-1) if x &lt; 0.<a name="line.688"></a>
<FONT color="green">689</FONT>         * <a name="line.689"></a>
<FONT color="green">690</FONT>         * @param x the value, a byte<a name="line.690"></a>
<FONT color="green">691</FONT>         * @return (byte)(+1) or (byte)(-1), depending on the sign of x<a name="line.691"></a>
<FONT color="green">692</FONT>         */<a name="line.692"></a>
<FONT color="green">693</FONT>        public static byte indicator(final byte x) {<a name="line.693"></a>
<FONT color="green">694</FONT>            return (x &gt;= ZB) ? PB : NB;<a name="line.694"></a>
<FONT color="green">695</FONT>        }<a name="line.695"></a>
<FONT color="green">696</FONT>    <a name="line.696"></a>
<FONT color="green">697</FONT>        /**<a name="line.697"></a>
<FONT color="green">698</FONT>         * For a double precision value x, this method returns +1.0 if x &gt;= 0 and<a name="line.698"></a>
<FONT color="green">699</FONT>         * -1.0 if x &lt; 0. Returns &lt;code&gt;NaN&lt;/code&gt; if &lt;code&gt;x&lt;/code&gt; is<a name="line.699"></a>
<FONT color="green">700</FONT>         * &lt;code&gt;NaN&lt;/code&gt;.<a name="line.700"></a>
<FONT color="green">701</FONT>         * <a name="line.701"></a>
<FONT color="green">702</FONT>         * @param x the value, a double<a name="line.702"></a>
<FONT color="green">703</FONT>         * @return +1.0 or -1.0, depending on the sign of x<a name="line.703"></a>
<FONT color="green">704</FONT>         */<a name="line.704"></a>
<FONT color="green">705</FONT>        public static double indicator(final double x) {<a name="line.705"></a>
<FONT color="green">706</FONT>            if (Double.isNaN(x)) {<a name="line.706"></a>
<FONT color="green">707</FONT>                return Double.NaN;<a name="line.707"></a>
<FONT color="green">708</FONT>            }<a name="line.708"></a>
<FONT color="green">709</FONT>            return (x &gt;= 0.0) ? 1.0 : -1.0;<a name="line.709"></a>
<FONT color="green">710</FONT>        }<a name="line.710"></a>
<FONT color="green">711</FONT>    <a name="line.711"></a>
<FONT color="green">712</FONT>        /**<a name="line.712"></a>
<FONT color="green">713</FONT>         * For a float value x, this method returns +1.0F if x &gt;= 0 and -1.0F if x &lt;<a name="line.713"></a>
<FONT color="green">714</FONT>         * 0. Returns &lt;code&gt;NaN&lt;/code&gt; if &lt;code&gt;x&lt;/code&gt; is &lt;code&gt;NaN&lt;/code&gt;.<a name="line.714"></a>
<FONT color="green">715</FONT>         * <a name="line.715"></a>
<FONT color="green">716</FONT>         * @param x the value, a float<a name="line.716"></a>
<FONT color="green">717</FONT>         * @return +1.0F or -1.0F, depending on the sign of x<a name="line.717"></a>
<FONT color="green">718</FONT>         */<a name="line.718"></a>
<FONT color="green">719</FONT>        public static float indicator(final float x) {<a name="line.719"></a>
<FONT color="green">720</FONT>            if (Float.isNaN(x)) {<a name="line.720"></a>
<FONT color="green">721</FONT>                return Float.NaN;<a name="line.721"></a>
<FONT color="green">722</FONT>            }<a name="line.722"></a>
<FONT color="green">723</FONT>            return (x &gt;= 0.0F) ? 1.0F : -1.0F;<a name="line.723"></a>
<FONT color="green">724</FONT>        }<a name="line.724"></a>
<FONT color="green">725</FONT>    <a name="line.725"></a>
<FONT color="green">726</FONT>        /**<a name="line.726"></a>
<FONT color="green">727</FONT>         * For an int value x, this method returns +1 if x &gt;= 0 and -1 if x &lt; 0.<a name="line.727"></a>
<FONT color="green">728</FONT>         * <a name="line.728"></a>
<FONT color="green">729</FONT>         * @param x the value, an int<a name="line.729"></a>
<FONT color="green">730</FONT>         * @return +1 or -1, depending on the sign of x<a name="line.730"></a>
<FONT color="green">731</FONT>         */<a name="line.731"></a>
<FONT color="green">732</FONT>        public static int indicator(final int x) {<a name="line.732"></a>
<FONT color="green">733</FONT>            return (x &gt;= 0) ? 1 : -1;<a name="line.733"></a>
<FONT color="green">734</FONT>        }<a name="line.734"></a>
<FONT color="green">735</FONT>    <a name="line.735"></a>
<FONT color="green">736</FONT>        /**<a name="line.736"></a>
<FONT color="green">737</FONT>         * For a long value x, this method returns +1L if x &gt;= 0 and -1L if x &lt; 0.<a name="line.737"></a>
<FONT color="green">738</FONT>         * <a name="line.738"></a>
<FONT color="green">739</FONT>         * @param x the value, a long<a name="line.739"></a>
<FONT color="green">740</FONT>         * @return +1L or -1L, depending on the sign of x<a name="line.740"></a>
<FONT color="green">741</FONT>         */<a name="line.741"></a>
<FONT color="green">742</FONT>        public static long indicator(final long x) {<a name="line.742"></a>
<FONT color="green">743</FONT>            return (x &gt;= 0L) ? 1L : -1L;<a name="line.743"></a>
<FONT color="green">744</FONT>        }<a name="line.744"></a>
<FONT color="green">745</FONT>    <a name="line.745"></a>
<FONT color="green">746</FONT>        /**<a name="line.746"></a>
<FONT color="green">747</FONT>         * For a short value x, this method returns (short)(+1) if x &gt;= 0 and<a name="line.747"></a>
<FONT color="green">748</FONT>         * (short)(-1) if x &lt; 0.<a name="line.748"></a>
<FONT color="green">749</FONT>         * <a name="line.749"></a>
<FONT color="green">750</FONT>         * @param x the value, a short<a name="line.750"></a>
<FONT color="green">751</FONT>         * @return (short)(+1) or (short)(-1), depending on the sign of x<a name="line.751"></a>
<FONT color="green">752</FONT>         */<a name="line.752"></a>
<FONT color="green">753</FONT>        public static short indicator(final short x) {<a name="line.753"></a>
<FONT color="green">754</FONT>            return (x &gt;= ZS) ? PS : NS;<a name="line.754"></a>
<FONT color="green">755</FONT>        }<a name="line.755"></a>
<FONT color="green">756</FONT>    <a name="line.756"></a>
<FONT color="green">757</FONT>        /**<a name="line.757"></a>
<FONT color="green">758</FONT>         * &lt;p&gt;<a name="line.758"></a>
<FONT color="green">759</FONT>         * Returns the least common multiple of the absolute value of two numbers,<a name="line.759"></a>
<FONT color="green">760</FONT>         * using the formula &lt;code&gt;lcm(a,b) = (a / gcd(a,b)) * b&lt;/code&gt;.<a name="line.760"></a>
<FONT color="green">761</FONT>         * &lt;/p&gt;<a name="line.761"></a>
<FONT color="green">762</FONT>         * Special cases:<a name="line.762"></a>
<FONT color="green">763</FONT>         * &lt;ul&gt;<a name="line.763"></a>
<FONT color="green">764</FONT>         * &lt;li&gt;The invocations &lt;code&gt;lcm(Integer.MIN_VALUE, n)&lt;/code&gt; and<a name="line.764"></a>
<FONT color="green">765</FONT>         * &lt;code&gt;lcm(n, Integer.MIN_VALUE)&lt;/code&gt;, where &lt;code&gt;abs(n)&lt;/code&gt; is a<a name="line.765"></a>
<FONT color="green">766</FONT>         * power of 2, throw an &lt;code&gt;ArithmeticException&lt;/code&gt;, because the result<a name="line.766"></a>
<FONT color="green">767</FONT>         * would be 2^31, which is too large for an int value.&lt;/li&gt;<a name="line.767"></a>
<FONT color="green">768</FONT>         * &lt;li&gt;The result of &lt;code&gt;lcm(0, x)&lt;/code&gt; and &lt;code&gt;lcm(x, 0)&lt;/code&gt; is<a name="line.768"></a>
<FONT color="green">769</FONT>         * &lt;code&gt;0&lt;/code&gt; for any &lt;code&gt;x&lt;/code&gt;.<a name="line.769"></a>
<FONT color="green">770</FONT>         * &lt;/ul&gt;<a name="line.770"></a>
<FONT color="green">771</FONT>         * <a name="line.771"></a>
<FONT color="green">772</FONT>         * @param a any number<a name="line.772"></a>
<FONT color="green">773</FONT>         * @param b any number<a name="line.773"></a>
<FONT color="green">774</FONT>         * @return the least common multiple, never negative<a name="line.774"></a>
<FONT color="green">775</FONT>         * @throws ArithmeticException<a name="line.775"></a>
<FONT color="green">776</FONT>         *             if the result cannot be represented as a nonnegative int<a name="line.776"></a>
<FONT color="green">777</FONT>         *             value<a name="line.777"></a>
<FONT color="green">778</FONT>         * @since 1.1<a name="line.778"></a>
<FONT color="green">779</FONT>         */<a name="line.779"></a>
<FONT color="green">780</FONT>        public static int lcm(int a, int b) {<a name="line.780"></a>
<FONT color="green">781</FONT>            if (a==0 || b==0){<a name="line.781"></a>
<FONT color="green">782</FONT>                return 0;<a name="line.782"></a>
<FONT color="green">783</FONT>            }<a name="line.783"></a>
<FONT color="green">784</FONT>            int lcm = Math.abs(mulAndCheck(a / gcd(a, b), b));<a name="line.784"></a>
<FONT color="green">785</FONT>            if (lcm == Integer.MIN_VALUE){<a name="line.785"></a>
<FONT color="green">786</FONT>                throw new ArithmeticException("overflow: lcm is 2^31");<a name="line.786"></a>
<FONT color="green">787</FONT>            }<a name="line.787"></a>
<FONT color="green">788</FONT>            return lcm;<a name="line.788"></a>
<FONT color="green">789</FONT>        }<a name="line.789"></a>
<FONT color="green">790</FONT>    <a name="line.790"></a>
<FONT color="green">791</FONT>        /** <a name="line.791"></a>
<FONT color="green">792</FONT>         * &lt;p&gt;Returns the <a name="line.792"></a>
<FONT color="green">793</FONT>         * &lt;a href="http://mathworld.wolfram.com/Logarithm.html"&gt;logarithm&lt;/a&gt;<a name="line.793"></a>
<FONT color="green">794</FONT>         * for base &lt;code&gt;b&lt;/code&gt; of &lt;code&gt;x&lt;/code&gt;.<a name="line.794"></a>
<FONT color="green">795</FONT>         * &lt;/p&gt;<a name="line.795"></a>
<FONT color="green">796</FONT>         * &lt;p&gt;Returns &lt;code&gt;NaN&lt;code&gt; if either argument is negative.  If <a name="line.796"></a>
<FONT color="green">797</FONT>         * &lt;code&gt;base&lt;/code&gt; is 0 and &lt;code&gt;x&lt;/code&gt; is positive, 0 is returned.<a name="line.797"></a>
<FONT color="green">798</FONT>         * If &lt;code&gt;base&lt;/code&gt; is positive and &lt;code&gt;x&lt;/code&gt; is 0, <a name="line.798"></a>
<FONT color="green">799</FONT>         * &lt;code&gt;Double.NEGATIVE_INFINITY&lt;/code&gt; is returned.  If both arguments<a name="line.799"></a>
<FONT color="green">800</FONT>         * are 0, the result is &lt;code&gt;NaN&lt;/code&gt;.&lt;/p&gt;<a name="line.800"></a>
<FONT color="green">801</FONT>         * <a name="line.801"></a>
<FONT color="green">802</FONT>         * @param base the base of the logarithm, must be greater than 0<a name="line.802"></a>
<FONT color="green">803</FONT>         * @param x argument, must be greater than 0<a name="line.803"></a>
<FONT color="green">804</FONT>         * @return the value of the logarithm - the number y such that base^y = x.<a name="line.804"></a>
<FONT color="green">805</FONT>         * @since 1.2<a name="line.805"></a>
<FONT color="green">806</FONT>         */ <a name="line.806"></a>
<FONT color="green">807</FONT>        public static double log(double base, double x) {<a name="line.807"></a>
<FONT color="green">808</FONT>            return Math.log(x)/Math.log(base);<a name="line.808"></a>
<FONT color="green">809</FONT>        }<a name="line.809"></a>
<FONT color="green">810</FONT>    <a name="line.810"></a>
<FONT color="green">811</FONT>        /**<a name="line.811"></a>
<FONT color="green">812</FONT>         * Multiply two integers, checking for overflow.<a name="line.812"></a>
<FONT color="green">813</FONT>         * <a name="line.813"></a>
<FONT color="green">814</FONT>         * @param x a factor<a name="line.814"></a>
<FONT color="green">815</FONT>         * @param y a factor<a name="line.815"></a>
<FONT color="green">816</FONT>         * @return the product &lt;code&gt;x*y&lt;/code&gt;<a name="line.816"></a>
<FONT color="green">817</FONT>         * @throws ArithmeticException if the result can not be represented as an<a name="line.817"></a>
<FONT color="green">818</FONT>         *         int<a name="line.818"></a>
<FONT color="green">819</FONT>         * @since 1.1<a name="line.819"></a>
<FONT color="green">820</FONT>         */<a name="line.820"></a>
<FONT color="green">821</FONT>        public static int mulAndCheck(int x, int y) {<a name="line.821"></a>
<FONT color="green">822</FONT>            long m = ((long)x) * ((long)y);<a name="line.822"></a>
<FONT color="green">823</FONT>            if (m &lt; Integer.MIN_VALUE || m &gt; Integer.MAX_VALUE) {<a name="line.823"></a>
<FONT color="green">824</FONT>                throw new ArithmeticException("overflow: mul");<a name="line.824"></a>
<FONT color="green">825</FONT>            }<a name="line.825"></a>
<FONT color="green">826</FONT>            return (int)m;<a name="line.826"></a>
<FONT color="green">827</FONT>        }<a name="line.827"></a>
<FONT color="green">828</FONT>    <a name="line.828"></a>
<FONT color="green">829</FONT>        /**<a name="line.829"></a>
<FONT color="green">830</FONT>         * Multiply two long integers, checking for overflow.<a name="line.830"></a>
<FONT color="green">831</FONT>         * <a name="line.831"></a>
<FONT color="green">832</FONT>         * @param a first value<a name="line.832"></a>
<FONT color="green">833</FONT>         * @param b second value<a name="line.833"></a>
<FONT color="green">834</FONT>         * @return the product &lt;code&gt;a * b&lt;/code&gt;<a name="line.834"></a>
<FONT color="green">835</FONT>         * @throws ArithmeticException if the result can not be represented as an<a name="line.835"></a>
<FONT color="green">836</FONT>         *         long<a name="line.836"></a>
<FONT color="green">837</FONT>         * @since 1.2<a name="line.837"></a>
<FONT color="green">838</FONT>         */<a name="line.838"></a>
<FONT color="green">839</FONT>        public static long mulAndCheck(long a, long b) {<a name="line.839"></a>
<FONT color="green">840</FONT>            long ret;<a name="line.840"></a>
<FONT color="green">841</FONT>            String msg = "overflow: multiply";<a name="line.841"></a>
<FONT color="green">842</FONT>            if (a &gt; b) {<a name="line.842"></a>
<FONT color="green">843</FONT>                // use symmetry to reduce boundary cases<a name="line.843"></a>
<FONT color="green">844</FONT>                ret = mulAndCheck(b, a);<a name="line.844"></a>
<FONT color="green">845</FONT>            } else {<a name="line.845"></a>
<FONT color="green">846</FONT>                if (a &lt; 0) {<a name="line.846"></a>
<FONT color="green">847</FONT>                    if (b &lt; 0) {<a name="line.847"></a>
<FONT color="green">848</FONT>                        // check for positive overflow with negative a, negative b<a name="line.848"></a>
<FONT color="green">849</FONT>                        if (a &gt;= Long.MAX_VALUE / b) {<a name="line.849"></a>
<FONT color="green">850</FONT>                            ret = a * b;<a name="line.850"></a>
<FONT color="green">851</FONT>                        } else {<a name="line.851"></a>
<FONT color="green">852</FONT>                            throw new ArithmeticException(msg);<a name="line.852"></a>
<FONT color="green">853</FONT>                        }<a name="line.853"></a>
<FONT color="green">854</FONT>                    } else if (b &gt; 0) {<a name="line.854"></a>
<FONT color="green">855</FONT>                        // check for negative overflow with negative a, positive b<a name="line.855"></a>
<FONT color="green">856</FONT>                        if (Long.MIN_VALUE / b &lt;= a) {<a name="line.856"></a>
<FONT color="green">857</FONT>                            ret = a * b;<a name="line.857"></a>
<FONT color="green">858</FONT>                        } else {<a name="line.858"></a>
<FONT color="green">859</FONT>                            throw new ArithmeticException(msg);<a name="line.859"></a>
<FONT color="green">860</FONT>                            <a name="line.860"></a>
<FONT color="green">861</FONT>                        }<a name="line.861"></a>
<FONT color="green">862</FONT>                    } else {<a name="line.862"></a>
<FONT color="green">863</FONT>                        // assert b == 0<a name="line.863"></a>
<FONT color="green">864</FONT>                        ret = 0;<a name="line.864"></a>
<FONT color="green">865</FONT>                    }<a name="line.865"></a>
<FONT color="green">866</FONT>                } else if (a &gt; 0) {<a name="line.866"></a>
<FONT color="green">867</FONT>                    // assert a &gt; 0<a name="line.867"></a>
<FONT color="green">868</FONT>                    // assert b &gt; 0<a name="line.868"></a>
<FONT color="green">869</FONT>                    <a name="line.869"></a>
<FONT color="green">870</FONT>                    // check for positive overflow with positive a, positive b<a name="line.870"></a>
<FONT color="green">871</FONT>                    if (a &lt;= Long.MAX_VALUE / b) {<a name="line.871"></a>
<FONT color="green">872</FONT>                        ret = a * b;<a name="line.872"></a>
<FONT color="green">873</FONT>                    } else {<a name="line.873"></a>
<FONT color="green">874</FONT>                        throw new ArithmeticException(msg);<a name="line.874"></a>
<FONT color="green">875</FONT>                    }<a name="line.875"></a>
<FONT color="green">876</FONT>                } else {<a name="line.876"></a>
<FONT color="green">877</FONT>                    // assert a == 0<a name="line.877"></a>
<FONT color="green">878</FONT>                    ret = 0;<a name="line.878"></a>
<FONT color="green">879</FONT>                }<a name="line.879"></a>
<FONT color="green">880</FONT>            }<a name="line.880"></a>
<FONT color="green">881</FONT>            return ret;<a name="line.881"></a>
<FONT color="green">882</FONT>        }<a name="line.882"></a>
<FONT color="green">883</FONT>    <a name="line.883"></a>
<FONT color="green">884</FONT>        /**<a name="line.884"></a>
<FONT color="green">885</FONT>         * Get the next machine representable number after a number, moving<a name="line.885"></a>
<FONT color="green">886</FONT>         * in the direction of another number.<a name="line.886"></a>
<FONT color="green">887</FONT>         * &lt;p&gt;<a name="line.887"></a>
<FONT color="green">888</FONT>         * If &lt;code&gt;direction&lt;/code&gt; is greater than or equal to&lt;code&gt;d&lt;/code&gt;,<a name="line.888"></a>
<FONT color="green">889</FONT>         * the smallest machine representable number strictly greater than<a name="line.889"></a>
<FONT color="green">890</FONT>         * &lt;code&gt;d&lt;/code&gt; is returned; otherwise the largest representable number<a name="line.890"></a>
<FONT color="green">891</FONT>         * strictly less than &lt;code&gt;d&lt;/code&gt; is returned.&lt;/p&gt;<a name="line.891"></a>
<FONT color="green">892</FONT>         * &lt;p&gt;<a name="line.892"></a>
<FONT color="green">893</FONT>         * If &lt;code&gt;d&lt;/code&gt; is NaN or Infinite, it is returned unchanged.&lt;/p&gt;<a name="line.893"></a>
<FONT color="green">894</FONT>         * <a name="line.894"></a>
<FONT color="green">895</FONT>         * @param d base number<a name="line.895"></a>
<FONT color="green">896</FONT>         * @param direction (the only important thing is whether<a name="line.896"></a>
<FONT color="green">897</FONT>         * direction is greater or smaller than d)<a name="line.897"></a>
<FONT color="green">898</FONT>         * @return the next machine representable number in the specified direction<a name="line.898"></a>
<FONT color="green">899</FONT>         * @since 1.2<a name="line.899"></a>
<FONT color="green">900</FONT>         */<a name="line.900"></a>
<FONT color="green">901</FONT>        public static double nextAfter(double d, double direction) {<a name="line.901"></a>
<FONT color="green">902</FONT>    <a name="line.902"></a>
<FONT color="green">903</FONT>            // handling of some important special cases<a name="line.903"></a>
<FONT color="green">904</FONT>            if (Double.isNaN(d) || Double.isInfinite(d)) {<a name="line.904"></a>
<FONT color="green">905</FONT>                    return d;<a name="line.905"></a>
<FONT color="green">906</FONT>            } else if (d == 0) {<a name="line.906"></a>
<FONT color="green">907</FONT>                    return (direction &lt; 0) ? -Double.MIN_VALUE : Double.MIN_VALUE;<a name="line.907"></a>
<FONT color="green">908</FONT>            }<a name="line.908"></a>
<FONT color="green">909</FONT>            // special cases MAX_VALUE to infinity and  MIN_VALUE to 0<a name="line.909"></a>
<FONT color="green">910</FONT>            // are handled just as normal numbers<a name="line.910"></a>
<FONT color="green">911</FONT>    <a name="line.911"></a>
<FONT color="green">912</FONT>            // split the double in raw components<a name="line.912"></a>
<FONT color="green">913</FONT>            long bits     = Double.doubleToLongBits(d);<a name="line.913"></a>
<FONT color="green">914</FONT>            long sign     = bits &amp; 0x8000000000000000L;<a name="line.914"></a>
<FONT color="green">915</FONT>            long exponent = bits &amp; 0x7ff0000000000000L;<a name="line.915"></a>
<FONT color="green">916</FONT>            long mantissa = bits &amp; 0x000fffffffffffffL;<a name="line.916"></a>
<FONT color="green">917</FONT>    <a name="line.917"></a>
<FONT color="green">918</FONT>            if (d * (direction - d) &gt;= 0) {<a name="line.918"></a>
<FONT color="green">919</FONT>                    // we should increase the mantissa<a name="line.919"></a>
<FONT color="green">920</FONT>                    if (mantissa == 0x000fffffffffffffL) {<a name="line.920"></a>
<FONT color="green">921</FONT>                            return Double.longBitsToDouble(sign |<a name="line.921"></a>
<FONT color="green">922</FONT>                                            (exponent + 0x0010000000000000L));<a name="line.922"></a>
<FONT color="green">923</FONT>                    } else {<a name="line.923"></a>
<FONT color="green">924</FONT>                            return Double.longBitsToDouble(sign |<a name="line.924"></a>
<FONT color="green">925</FONT>                                            exponent | (mantissa + 1));<a name="line.925"></a>
<FONT color="green">926</FONT>                    }<a name="line.926"></a>
<FONT color="green">927</FONT>            } else {<a name="line.927"></a>
<FONT color="green">928</FONT>                    // we should decrease the mantissa<a name="line.928"></a>
<FONT color="green">929</FONT>                    if (mantissa == 0L) {<a name="line.929"></a>
<FONT color="green">930</FONT>                            return Double.longBitsToDouble(sign |<a name="line.930"></a>
<FONT color="green">931</FONT>                                            (exponent - 0x0010000000000000L) |<a name="line.931"></a>
<FONT color="green">932</FONT>                                            0x000fffffffffffffL);<a name="line.932"></a>
<FONT color="green">933</FONT>                    } else {<a name="line.933"></a>
<FONT color="green">934</FONT>                            return Double.longBitsToDouble(sign |<a name="line.934"></a>
<FONT color="green">935</FONT>                                            exponent | (mantissa - 1));<a name="line.935"></a>
<FONT color="green">936</FONT>                    }<a name="line.936"></a>
<FONT color="green">937</FONT>            }<a name="line.937"></a>
<FONT color="green">938</FONT>    <a name="line.938"></a>
<FONT color="green">939</FONT>        }<a name="line.939"></a>
<FONT color="green">940</FONT>    <a name="line.940"></a>
<FONT color="green">941</FONT>        /**<a name="line.941"></a>
<FONT color="green">942</FONT>         * Scale a number by 2&lt;sup&gt;scaleFactor&lt;/sup&gt;.<a name="line.942"></a>
<FONT color="green">943</FONT>         * &lt;p&gt;If &lt;code&gt;d&lt;/code&gt; is 0 or NaN or Infinite, it is returned unchanged.&lt;/p&gt;<a name="line.943"></a>
<FONT color="green">944</FONT>         * <a name="line.944"></a>
<FONT color="green">945</FONT>         * @param d base number<a name="line.945"></a>
<FONT color="green">946</FONT>         * @param scaleFactor power of two by which d sould be multiplied<a name="line.946"></a>
<FONT color="green">947</FONT>         * @return d &amp;times; 2&lt;sup&gt;scaleFactor&lt;/sup&gt;<a name="line.947"></a>
<FONT color="green">948</FONT>         * @since 2.0<a name="line.948"></a>
<FONT color="green">949</FONT>         */<a name="line.949"></a>
<FONT color="green">950</FONT>        public static double scalb(final double d, final int scaleFactor) {<a name="line.950"></a>
<FONT color="green">951</FONT>    <a name="line.951"></a>
<FONT color="green">952</FONT>            // handling of some important special cases<a name="line.952"></a>
<FONT color="green">953</FONT>            if ((d == 0) || Double.isNaN(d) || Double.isInfinite(d)) {<a name="line.953"></a>
<FONT color="green">954</FONT>                return d;<a name="line.954"></a>
<FONT color="green">955</FONT>            }<a name="line.955"></a>
<FONT color="green">956</FONT>    <a name="line.956"></a>
<FONT color="green">957</FONT>            // split the double in raw components<a name="line.957"></a>
<FONT color="green">958</FONT>            final long bits     = Double.doubleToLongBits(d);<a name="line.958"></a>
<FONT color="green">959</FONT>            final long exponent = bits &amp; 0x7ff0000000000000L;<a name="line.959"></a>
<FONT color="green">960</FONT>            final long rest     = bits &amp; 0x800fffffffffffffL;<a name="line.960"></a>
<FONT color="green">961</FONT>    <a name="line.961"></a>
<FONT color="green">962</FONT>            // shift the exponent<a name="line.962"></a>
<FONT color="green">963</FONT>            final long newBits = rest | (exponent + (((long) scaleFactor) &lt;&lt; 52));<a name="line.963"></a>
<FONT color="green">964</FONT>            return Double.longBitsToDouble(newBits);<a name="line.964"></a>
<FONT color="green">965</FONT>    <a name="line.965"></a>
<FONT color="green">966</FONT>        }<a name="line.966"></a>
<FONT color="green">967</FONT>    <a name="line.967"></a>
<FONT color="green">968</FONT>        /**<a name="line.968"></a>
<FONT color="green">969</FONT>         * Normalize an angle in a 2&amp;pi wide interval around a center value.<a name="line.969"></a>
<FONT color="green">970</FONT>         * &lt;p&gt;This method has three main uses:&lt;/p&gt;<a name="line.970"></a>
<FONT color="green">971</FONT>         * &lt;ul&gt;<a name="line.971"></a>
<FONT color="green">972</FONT>         *   &lt;li&gt;normalize an angle between 0 and 2&amp;pi;:&lt;br/&gt;<a name="line.972"></a>
<FONT color="green">973</FONT>         *       &lt;code&gt;a = MathUtils.normalizeAngle(a, Math.PI);&lt;/code&gt;&lt;/li&gt;<a name="line.973"></a>
<FONT color="green">974</FONT>         *   &lt;li&gt;normalize an angle between -&amp;pi; and +&amp;pi;&lt;br/&gt;<a name="line.974"></a>
<FONT color="green">975</FONT>         *       &lt;code&gt;a = MathUtils.normalizeAngle(a, 0.0);&lt;/code&gt;&lt;/li&gt;<a name="line.975"></a>
<FONT color="green">976</FONT>         *   &lt;li&gt;compute the angle between two defining angular positions:&lt;br&gt;<a name="line.976"></a>
<FONT color="green">977</FONT>         *       &lt;code&gt;angle = MathUtils.normalizeAngle(end, start) - start;&lt;/code&gt;&lt;/li&gt;<a name="line.977"></a>
<FONT color="green">978</FONT>         * &lt;/ul&gt;<a name="line.978"></a>
<FONT color="green">979</FONT>         * &lt;p&gt;Note that due to numerical accuracy and since &amp;pi; cannot be represented<a name="line.979"></a>
<FONT color="green">980</FONT>         * exactly, the result interval is &lt;em&gt;closed&lt;/em&gt;, it cannot be half-closed<a name="line.980"></a>
<FONT color="green">981</FONT>         * as would be more satisfactory in a purely mathematical view.&lt;/p&gt;<a name="line.981"></a>
<FONT color="green">982</FONT>         * @param a angle to normalize<a name="line.982"></a>
<FONT color="green">983</FONT>         * @param center center of the desired 2&amp;pi; interval for the result<a name="line.983"></a>
<FONT color="green">984</FONT>         * @return a-2k&amp;pi; with integer k and center-&amp;pi; &amp;lt;= a-2k&amp;pi; &amp;lt;= center+&amp;pi;<a name="line.984"></a>
<FONT color="green">985</FONT>         * @since 1.2<a name="line.985"></a>
<FONT color="green">986</FONT>         */<a name="line.986"></a>
<FONT color="green">987</FONT>         public static double normalizeAngle(double a, double center) {<a name="line.987"></a>
<FONT color="green">988</FONT>             return a - TWO_PI * Math.floor((a + Math.PI - center) / TWO_PI);<a name="line.988"></a>
<FONT color="green">989</FONT>         }<a name="line.989"></a>
<FONT color="green">990</FONT>    <a name="line.990"></a>
<FONT color="green">991</FONT>        /**<a name="line.991"></a>
<FONT color="green">992</FONT>         * Round the given value to the specified number of decimal places. The<a name="line.992"></a>
<FONT color="green">993</FONT>         * value is rounded using the {@link BigDecimal#ROUND_HALF_UP} method.<a name="line.993"></a>
<FONT color="green">994</FONT>         * <a name="line.994"></a>
<FONT color="green">995</FONT>         * @param x the value to round.<a name="line.995"></a>
<FONT color="green">996</FONT>         * @param scale the number of digits to the right of the decimal point.<a name="line.996"></a>
<FONT color="green">997</FONT>         * @return the rounded value.<a name="line.997"></a>
<FONT color="green">998</FONT>         * @since 1.1<a name="line.998"></a>
<FONT color="green">999</FONT>         */<a name="line.999"></a>
<FONT color="green">1000</FONT>        public static double round(double x, int scale) {<a name="line.1000"></a>
<FONT color="green">1001</FONT>            return round(x, scale, BigDecimal.ROUND_HALF_UP);<a name="line.1001"></a>
<FONT color="green">1002</FONT>        }<a name="line.1002"></a>
<FONT color="green">1003</FONT>    <a name="line.1003"></a>
<FONT color="green">1004</FONT>        /**<a name="line.1004"></a>
<FONT color="green">1005</FONT>         * Round the given value to the specified number of decimal places. The<a name="line.1005"></a>
<FONT color="green">1006</FONT>         * value is rounded using the given method which is any method defined in<a name="line.1006"></a>
<FONT color="green">1007</FONT>         * {@link BigDecimal}.<a name="line.1007"></a>
<FONT color="green">1008</FONT>         * <a name="line.1008"></a>
<FONT color="green">1009</FONT>         * @param x the value to round.<a name="line.1009"></a>
<FONT color="green">1010</FONT>         * @param scale the number of digits to the right of the decimal point.<a name="line.1010"></a>
<FONT color="green">1011</FONT>         * @param roundingMethod the rounding method as defined in<a name="line.1011"></a>
<FONT color="green">1012</FONT>         *        {@link BigDecimal}.<a name="line.1012"></a>
<FONT color="green">1013</FONT>         * @return the rounded value.<a name="line.1013"></a>
<FONT color="green">1014</FONT>         * @since 1.1<a name="line.1014"></a>
<FONT color="green">1015</FONT>         */<a name="line.1015"></a>
<FONT color="green">1016</FONT>        public static double round(double x, int scale, int roundingMethod) {<a name="line.1016"></a>
<FONT color="green">1017</FONT>            try {<a name="line.1017"></a>
<FONT color="green">1018</FONT>                return (new BigDecimal<a name="line.1018"></a>
<FONT color="green">1019</FONT>                       (Double.toString(x))<a name="line.1019"></a>
<FONT color="green">1020</FONT>                       .setScale(scale, roundingMethod))<a name="line.1020"></a>
<FONT color="green">1021</FONT>                       .doubleValue();<a name="line.1021"></a>
<FONT color="green">1022</FONT>            } catch (NumberFormatException ex) {<a name="line.1022"></a>
<FONT color="green">1023</FONT>                if (Double.isInfinite(x)) {<a name="line.1023"></a>
<FONT color="green">1024</FONT>                    return x;          <a name="line.1024"></a>
<FONT color="green">1025</FONT>                } else {<a name="line.1025"></a>
<FONT color="green">1026</FONT>                    return Double.NaN;<a name="line.1026"></a>
<FONT color="green">1027</FONT>                }<a name="line.1027"></a>
<FONT color="green">1028</FONT>            }<a name="line.1028"></a>
<FONT color="green">1029</FONT>        }<a name="line.1029"></a>
<FONT color="green">1030</FONT>    <a name="line.1030"></a>
<FONT color="green">1031</FONT>        /**<a name="line.1031"></a>
<FONT color="green">1032</FONT>         * Round the given value to the specified number of decimal places. The<a name="line.1032"></a>
<FONT color="green">1033</FONT>         * value is rounding using the {@link BigDecimal#ROUND_HALF_UP} method.<a name="line.1033"></a>
<FONT color="green">1034</FONT>         * <a name="line.1034"></a>
<FONT color="green">1035</FONT>         * @param x the value to round.<a name="line.1035"></a>
<FONT color="green">1036</FONT>         * @param scale the number of digits to the right of the decimal point.<a name="line.1036"></a>
<FONT color="green">1037</FONT>         * @return the rounded value.<a name="line.1037"></a>
<FONT color="green">1038</FONT>         * @since 1.1<a name="line.1038"></a>
<FONT color="green">1039</FONT>         */<a name="line.1039"></a>
<FONT color="green">1040</FONT>        public static float round(float x, int scale) {<a name="line.1040"></a>
<FONT color="green">1041</FONT>            return round(x, scale, BigDecimal.ROUND_HALF_UP);<a name="line.1041"></a>
<FONT color="green">1042</FONT>        }<a name="line.1042"></a>
<FONT color="green">1043</FONT>    <a name="line.1043"></a>
<FONT color="green">1044</FONT>        /**<a name="line.1044"></a>
<FONT color="green">1045</FONT>         * Round the given value to the specified number of decimal places. The<a name="line.1045"></a>
<FONT color="green">1046</FONT>         * value is rounded using the given method which is any method defined in<a name="line.1046"></a>
<FONT color="green">1047</FONT>         * {@link BigDecimal}.<a name="line.1047"></a>
<FONT color="green">1048</FONT>         * <a name="line.1048"></a>
<FONT color="green">1049</FONT>         * @param x the value to round.<a name="line.1049"></a>
<FONT color="green">1050</FONT>         * @param scale the number of digits to the right of the decimal point.<a name="line.1050"></a>
<FONT color="green">1051</FONT>         * @param roundingMethod the rounding method as defined in<a name="line.1051"></a>
<FONT color="green">1052</FONT>         *        {@link BigDecimal}.<a name="line.1052"></a>
<FONT color="green">1053</FONT>         * @return the rounded value.<a name="line.1053"></a>
<FONT color="green">1054</FONT>         * @since 1.1<a name="line.1054"></a>
<FONT color="green">1055</FONT>         */<a name="line.1055"></a>
<FONT color="green">1056</FONT>        public static float round(float x, int scale, int roundingMethod) {<a name="line.1056"></a>
<FONT color="green">1057</FONT>            float sign = indicator(x);<a name="line.1057"></a>
<FONT color="green">1058</FONT>            float factor = (float)Math.pow(10.0f, scale) * sign;<a name="line.1058"></a>
<FONT color="green">1059</FONT>            return (float)roundUnscaled(x * factor, sign, roundingMethod) / factor;<a name="line.1059"></a>
<FONT color="green">1060</FONT>        }<a name="line.1060"></a>
<FONT color="green">1061</FONT>    <a name="line.1061"></a>
<FONT color="green">1062</FONT>        /**<a name="line.1062"></a>
<FONT color="green">1063</FONT>         * Round the given non-negative, value to the "nearest" integer. Nearest is<a name="line.1063"></a>
<FONT color="green">1064</FONT>         * determined by the rounding method specified. Rounding methods are defined<a name="line.1064"></a>
<FONT color="green">1065</FONT>         * in {@link BigDecimal}.<a name="line.1065"></a>
<FONT color="green">1066</FONT>         * <a name="line.1066"></a>
<FONT color="green">1067</FONT>         * @param unscaled the value to round.<a name="line.1067"></a>
<FONT color="green">1068</FONT>         * @param sign the sign of the original, scaled value.<a name="line.1068"></a>
<FONT color="green">1069</FONT>         * @param roundingMethod the rounding method as defined in<a name="line.1069"></a>
<FONT color="green">1070</FONT>         *        {@link BigDecimal}.<a name="line.1070"></a>
<FONT color="green">1071</FONT>         * @return the rounded value.<a name="line.1071"></a>
<FONT color="green">1072</FONT>         * @since 1.1<a name="line.1072"></a>
<FONT color="green">1073</FONT>         */<a name="line.1073"></a>
<FONT color="green">1074</FONT>        private static double roundUnscaled(double unscaled, double sign,<a name="line.1074"></a>
<FONT color="green">1075</FONT>            int roundingMethod) {<a name="line.1075"></a>
<FONT color="green">1076</FONT>            switch (roundingMethod) {<a name="line.1076"></a>
<FONT color="green">1077</FONT>            case BigDecimal.ROUND_CEILING :<a name="line.1077"></a>
<FONT color="green">1078</FONT>                if (sign == -1) {<a name="line.1078"></a>
<FONT color="green">1079</FONT>                    unscaled = Math.floor(nextAfter(unscaled, Double.NEGATIVE_INFINITY));<a name="line.1079"></a>
<FONT color="green">1080</FONT>                } else {<a name="line.1080"></a>
<FONT color="green">1081</FONT>                    unscaled = Math.ceil(nextAfter(unscaled, Double.POSITIVE_INFINITY));<a name="line.1081"></a>
<FONT color="green">1082</FONT>                }<a name="line.1082"></a>
<FONT color="green">1083</FONT>                break;<a name="line.1083"></a>
<FONT color="green">1084</FONT>            case BigDecimal.ROUND_DOWN :<a name="line.1084"></a>
<FONT color="green">1085</FONT>                unscaled = Math.floor(nextAfter(unscaled, Double.NEGATIVE_INFINITY));<a name="line.1085"></a>
<FONT color="green">1086</FONT>                break;<a name="line.1086"></a>
<FONT color="green">1087</FONT>            case BigDecimal.ROUND_FLOOR :<a name="line.1087"></a>
<FONT color="green">1088</FONT>                if (sign == -1) {<a name="line.1088"></a>
<FONT color="green">1089</FONT>                    unscaled = Math.ceil(nextAfter(unscaled, Double.POSITIVE_INFINITY));<a name="line.1089"></a>
<FONT color="green">1090</FONT>                } else {<a name="line.1090"></a>
<FONT color="green">1091</FONT>                    unscaled = Math.floor(nextAfter(unscaled, Double.NEGATIVE_INFINITY));<a name="line.1091"></a>
<FONT color="green">1092</FONT>                }<a name="line.1092"></a>
<FONT color="green">1093</FONT>                break;<a name="line.1093"></a>
<FONT color="green">1094</FONT>            case BigDecimal.ROUND_HALF_DOWN : {<a name="line.1094"></a>
<FONT color="green">1095</FONT>                unscaled = nextAfter(unscaled, Double.NEGATIVE_INFINITY);<a name="line.1095"></a>
<FONT color="green">1096</FONT>                double fraction = unscaled - Math.floor(unscaled);<a name="line.1096"></a>
<FONT color="green">1097</FONT>                if (fraction &gt; 0.5) {<a name="line.1097"></a>
<FONT color="green">1098</FONT>                    unscaled = Math.ceil(unscaled);<a name="line.1098"></a>
<FONT color="green">1099</FONT>                } else {<a name="line.1099"></a>
<FONT color="green">1100</FONT>                    unscaled = Math.floor(unscaled);<a name="line.1100"></a>
<FONT color="green">1101</FONT>                }<a name="line.1101"></a>
<FONT color="green">1102</FONT>                break;<a name="line.1102"></a>
<FONT color="green">1103</FONT>            }<a name="line.1103"></a>
<FONT color="green">1104</FONT>            case BigDecimal.ROUND_HALF_EVEN : {<a name="line.1104"></a>
<FONT color="green">1105</FONT>                double fraction = unscaled - Math.floor(unscaled);<a name="line.1105"></a>
<FONT color="green">1106</FONT>                if (fraction &gt; 0.5) {<a name="line.1106"></a>
<FONT color="green">1107</FONT>                    unscaled = Math.ceil(unscaled);<a name="line.1107"></a>
<FONT color="green">1108</FONT>                } else if (fraction &lt; 0.5) {<a name="line.1108"></a>
<FONT color="green">1109</FONT>                    unscaled = Math.floor(unscaled);<a name="line.1109"></a>
<FONT color="green">1110</FONT>                } else {<a name="line.1110"></a>
<FONT color="green">1111</FONT>                    // The following equality test is intentional and needed for rounding purposes<a name="line.1111"></a>
<FONT color="green">1112</FONT>                    if (Math.floor(unscaled) / 2.0 == Math.floor(Math<a name="line.1112"></a>
<FONT color="green">1113</FONT>                        .floor(unscaled) / 2.0)) { // even<a name="line.1113"></a>
<FONT color="green">1114</FONT>                        unscaled = Math.floor(unscaled);<a name="line.1114"></a>
<FONT color="green">1115</FONT>                    } else { // odd<a name="line.1115"></a>
<FONT color="green">1116</FONT>                        unscaled = Math.ceil(unscaled);<a name="line.1116"></a>
<FONT color="green">1117</FONT>                    }<a name="line.1117"></a>
<FONT color="green">1118</FONT>                }<a name="line.1118"></a>
<FONT color="green">1119</FONT>                break;<a name="line.1119"></a>
<FONT color="green">1120</FONT>            }<a name="line.1120"></a>
<FONT color="green">1121</FONT>            case BigDecimal.ROUND_HALF_UP : {<a name="line.1121"></a>
<FONT color="green">1122</FONT>                unscaled = nextAfter(unscaled, Double.POSITIVE_INFINITY);<a name="line.1122"></a>
<FONT color="green">1123</FONT>                double fraction = unscaled - Math.floor(unscaled);<a name="line.1123"></a>
<FONT color="green">1124</FONT>                if (fraction &gt;= 0.5) {<a name="line.1124"></a>
<FONT color="green">1125</FONT>                    unscaled = Math.ceil(unscaled);<a name="line.1125"></a>
<FONT color="green">1126</FONT>                } else {<a name="line.1126"></a>
<FONT color="green">1127</FONT>                    unscaled = Math.floor(unscaled);<a name="line.1127"></a>
<FONT color="green">1128</FONT>                }<a name="line.1128"></a>
<FONT color="green">1129</FONT>                break;<a name="line.1129"></a>
<FONT color="green">1130</FONT>            }<a name="line.1130"></a>
<FONT color="green">1131</FONT>            case BigDecimal.ROUND_UNNECESSARY :<a name="line.1131"></a>
<FONT color="green">1132</FONT>                if (unscaled != Math.floor(unscaled)) {<a name="line.1132"></a>
<FONT color="green">1133</FONT>                    throw new ArithmeticException("Inexact result from rounding");<a name="line.1133"></a>
<FONT color="green">1134</FONT>                }<a name="line.1134"></a>
<FONT color="green">1135</FONT>                break;<a name="line.1135"></a>
<FONT color="green">1136</FONT>            case BigDecimal.ROUND_UP :<a name="line.1136"></a>
<FONT color="green">1137</FONT>                unscaled = Math.ceil(nextAfter(unscaled,  Double.POSITIVE_INFINITY));<a name="line.1137"></a>
<FONT color="green">1138</FONT>                break;<a name="line.1138"></a>
<FONT color="green">1139</FONT>            default :<a name="line.1139"></a>
<FONT color="green">1140</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1140"></a>
<FONT color="green">1141</FONT>                      "invalid rounding method {0}, valid methods: {1} ({2}), {3} ({4})," +<a name="line.1141"></a>
<FONT color="green">1142</FONT>                      " {5} ({6}), {7} ({8}), {9} ({10}), {11} ({12}), {13} ({14}), {15} ({16})",<a name="line.1142"></a>
<FONT color="green">1143</FONT>                      roundingMethod,<a name="line.1143"></a>
<FONT color="green">1144</FONT>                      "ROUND_CEILING",     BigDecimal.ROUND_CEILING,<a name="line.1144"></a>
<FONT color="green">1145</FONT>                      "ROUND_DOWN",        BigDecimal.ROUND_DOWN,<a name="line.1145"></a>
<FONT color="green">1146</FONT>                      "ROUND_FLOOR",       BigDecimal.ROUND_FLOOR,<a name="line.1146"></a>
<FONT color="green">1147</FONT>                      "ROUND_HALF_DOWN",   BigDecimal.ROUND_HALF_DOWN,<a name="line.1147"></a>
<FONT color="green">1148</FONT>                      "ROUND_HALF_EVEN",   BigDecimal.ROUND_HALF_EVEN,<a name="line.1148"></a>
<FONT color="green">1149</FONT>                      "ROUND_HALF_UP",     BigDecimal.ROUND_HALF_UP,<a name="line.1149"></a>
<FONT color="green">1150</FONT>                      "ROUND_UNNECESSARY", BigDecimal.ROUND_UNNECESSARY,<a name="line.1150"></a>
<FONT color="green">1151</FONT>                      "ROUND_UP",          BigDecimal.ROUND_UP);<a name="line.1151"></a>
<FONT color="green">1152</FONT>            }<a name="line.1152"></a>
<FONT color="green">1153</FONT>            return unscaled;<a name="line.1153"></a>
<FONT color="green">1154</FONT>        }<a name="line.1154"></a>
<FONT color="green">1155</FONT>    <a name="line.1155"></a>
<FONT color="green">1156</FONT>        /**<a name="line.1156"></a>
<FONT color="green">1157</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/Sign.html"&gt; sign&lt;/a&gt;<a name="line.1157"></a>
<FONT color="green">1158</FONT>         * for byte value &lt;code&gt;x&lt;/code&gt;.<a name="line.1158"></a>
<FONT color="green">1159</FONT>         * &lt;p&gt;<a name="line.1159"></a>
<FONT color="green">1160</FONT>         * For a byte value x, this method returns (byte)(+1) if x &gt; 0, (byte)(0) if<a name="line.1160"></a>
<FONT color="green">1161</FONT>         * x = 0, and (byte)(-1) if x &lt; 0.&lt;/p&gt;<a name="line.1161"></a>
<FONT color="green">1162</FONT>         * <a name="line.1162"></a>
<FONT color="green">1163</FONT>         * @param x the value, a byte<a name="line.1163"></a>
<FONT color="green">1164</FONT>         * @return (byte)(+1), (byte)(0), or (byte)(-1), depending on the sign of x<a name="line.1164"></a>
<FONT color="green">1165</FONT>         */<a name="line.1165"></a>
<FONT color="green">1166</FONT>        public static byte sign(final byte x) {<a name="line.1166"></a>
<FONT color="green">1167</FONT>            return (x == ZB) ? ZB : (x &gt; ZB) ? PB : NB;<a name="line.1167"></a>
<FONT color="green">1168</FONT>        }<a name="line.1168"></a>
<FONT color="green">1169</FONT>    <a name="line.1169"></a>
<FONT color="green">1170</FONT>        /**<a name="line.1170"></a>
<FONT color="green">1171</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/Sign.html"&gt; sign&lt;/a&gt;<a name="line.1171"></a>
<FONT color="green">1172</FONT>         * for double precision &lt;code&gt;x&lt;/code&gt;.<a name="line.1172"></a>
<FONT color="green">1173</FONT>         * &lt;p&gt;<a name="line.1173"></a>
<FONT color="green">1174</FONT>         * For a double value &lt;code&gt;x&lt;/code&gt;, this method returns<a name="line.1174"></a>
<FONT color="green">1175</FONT>         * &lt;code&gt;+1.0&lt;/code&gt; if &lt;code&gt;x &gt; 0&lt;/code&gt;, &lt;code&gt;0.0&lt;/code&gt; if<a name="line.1175"></a>
<FONT color="green">1176</FONT>         * &lt;code&gt;x = 0.0&lt;/code&gt;, and &lt;code&gt;-1.0&lt;/code&gt; if &lt;code&gt;x &lt; 0&lt;/code&gt;.<a name="line.1176"></a>
<FONT color="green">1177</FONT>         * Returns &lt;code&gt;NaN&lt;/code&gt; if &lt;code&gt;x&lt;/code&gt; is &lt;code&gt;NaN&lt;/code&gt;.&lt;/p&gt;<a name="line.1177"></a>
<FONT color="green">1178</FONT>         * <a name="line.1178"></a>
<FONT color="green">1179</FONT>         * @param x the value, a double<a name="line.1179"></a>
<FONT color="green">1180</FONT>         * @return +1.0, 0.0, or -1.0, depending on the sign of x<a name="line.1180"></a>
<FONT color="green">1181</FONT>         */<a name="line.1181"></a>
<FONT color="green">1182</FONT>        public static double sign(final double x) {<a name="line.1182"></a>
<FONT color="green">1183</FONT>            if (Double.isNaN(x)) {<a name="line.1183"></a>
<FONT color="green">1184</FONT>                return Double.NaN;<a name="line.1184"></a>
<FONT color="green">1185</FONT>            }<a name="line.1185"></a>
<FONT color="green">1186</FONT>            return (x == 0.0) ? 0.0 : (x &gt; 0.0) ? 1.0 : -1.0;<a name="line.1186"></a>
<FONT color="green">1187</FONT>        }<a name="line.1187"></a>
<FONT color="green">1188</FONT>    <a name="line.1188"></a>
<FONT color="green">1189</FONT>        /**<a name="line.1189"></a>
<FONT color="green">1190</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/Sign.html"&gt; sign&lt;/a&gt;<a name="line.1190"></a>
<FONT color="green">1191</FONT>         * for float value &lt;code&gt;x&lt;/code&gt;.<a name="line.1191"></a>
<FONT color="green">1192</FONT>         * &lt;p&gt;<a name="line.1192"></a>
<FONT color="green">1193</FONT>         * For a float value x, this method returns +1.0F if x &gt; 0, 0.0F if x =<a name="line.1193"></a>
<FONT color="green">1194</FONT>         * 0.0F, and -1.0F if x &lt; 0. Returns &lt;code&gt;NaN&lt;/code&gt; if &lt;code&gt;x&lt;/code&gt;<a name="line.1194"></a>
<FONT color="green">1195</FONT>         * is &lt;code&gt;NaN&lt;/code&gt;.&lt;/p&gt;<a name="line.1195"></a>
<FONT color="green">1196</FONT>         * <a name="line.1196"></a>
<FONT color="green">1197</FONT>         * @param x the value, a float<a name="line.1197"></a>
<FONT color="green">1198</FONT>         * @return +1.0F, 0.0F, or -1.0F, depending on the sign of x<a name="line.1198"></a>
<FONT color="green">1199</FONT>         */<a name="line.1199"></a>
<FONT color="green">1200</FONT>        public static float sign(final float x) {<a name="line.1200"></a>
<FONT color="green">1201</FONT>            if (Float.isNaN(x)) {<a name="line.1201"></a>
<FONT color="green">1202</FONT>                return Float.NaN;<a name="line.1202"></a>
<FONT color="green">1203</FONT>            }<a name="line.1203"></a>
<FONT color="green">1204</FONT>            return (x == 0.0F) ? 0.0F : (x &gt; 0.0F) ? 1.0F : -1.0F;<a name="line.1204"></a>
<FONT color="green">1205</FONT>        }<a name="line.1205"></a>
<FONT color="green">1206</FONT>    <a name="line.1206"></a>
<FONT color="green">1207</FONT>        /**<a name="line.1207"></a>
<FONT color="green">1208</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/Sign.html"&gt; sign&lt;/a&gt;<a name="line.1208"></a>
<FONT color="green">1209</FONT>         * for int value &lt;code&gt;x&lt;/code&gt;.<a name="line.1209"></a>
<FONT color="green">1210</FONT>         * &lt;p&gt;<a name="line.1210"></a>
<FONT color="green">1211</FONT>         * For an int value x, this method returns +1 if x &gt; 0, 0 if x = 0, and -1<a name="line.1211"></a>
<FONT color="green">1212</FONT>         * if x &lt; 0.&lt;/p&gt;<a name="line.1212"></a>
<FONT color="green">1213</FONT>         * <a name="line.1213"></a>
<FONT color="green">1214</FONT>         * @param x the value, an int<a name="line.1214"></a>
<FONT color="green">1215</FONT>         * @return +1, 0, or -1, depending on the sign of x<a name="line.1215"></a>
<FONT color="green">1216</FONT>         */<a name="line.1216"></a>
<FONT color="green">1217</FONT>        public static int sign(final int x) {<a name="line.1217"></a>
<FONT color="green">1218</FONT>            return (x == 0) ? 0 : (x &gt; 0) ? 1 : -1;<a name="line.1218"></a>
<FONT color="green">1219</FONT>        }<a name="line.1219"></a>
<FONT color="green">1220</FONT>    <a name="line.1220"></a>
<FONT color="green">1221</FONT>        /**<a name="line.1221"></a>
<FONT color="green">1222</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/Sign.html"&gt; sign&lt;/a&gt;<a name="line.1222"></a>
<FONT color="green">1223</FONT>         * for long value &lt;code&gt;x&lt;/code&gt;.<a name="line.1223"></a>
<FONT color="green">1224</FONT>         * &lt;p&gt;<a name="line.1224"></a>
<FONT color="green">1225</FONT>         * For a long value x, this method returns +1L if x &gt; 0, 0L if x = 0, and<a name="line.1225"></a>
<FONT color="green">1226</FONT>         * -1L if x &lt; 0.&lt;/p&gt;<a name="line.1226"></a>
<FONT color="green">1227</FONT>         * <a name="line.1227"></a>
<FONT color="green">1228</FONT>         * @param x the value, a long<a name="line.1228"></a>
<FONT color="green">1229</FONT>         * @return +1L, 0L, or -1L, depending on the sign of x<a name="line.1229"></a>
<FONT color="green">1230</FONT>         */<a name="line.1230"></a>
<FONT color="green">1231</FONT>        public static long sign(final long x) {<a name="line.1231"></a>
<FONT color="green">1232</FONT>            return (x == 0L) ? 0L : (x &gt; 0L) ? 1L : -1L;<a name="line.1232"></a>
<FONT color="green">1233</FONT>        }<a name="line.1233"></a>
<FONT color="green">1234</FONT>    <a name="line.1234"></a>
<FONT color="green">1235</FONT>        /**<a name="line.1235"></a>
<FONT color="green">1236</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/Sign.html"&gt; sign&lt;/a&gt;<a name="line.1236"></a>
<FONT color="green">1237</FONT>         * for short value &lt;code&gt;x&lt;/code&gt;.<a name="line.1237"></a>
<FONT color="green">1238</FONT>         * &lt;p&gt;<a name="line.1238"></a>
<FONT color="green">1239</FONT>         * For a short value x, this method returns (short)(+1) if x &gt; 0, (short)(0)<a name="line.1239"></a>
<FONT color="green">1240</FONT>         * if x = 0, and (short)(-1) if x &lt; 0.&lt;/p&gt;<a name="line.1240"></a>
<FONT color="green">1241</FONT>         * <a name="line.1241"></a>
<FONT color="green">1242</FONT>         * @param x the value, a short<a name="line.1242"></a>
<FONT color="green">1243</FONT>         * @return (short)(+1), (short)(0), or (short)(-1), depending on the sign of<a name="line.1243"></a>
<FONT color="green">1244</FONT>         *         x<a name="line.1244"></a>
<FONT color="green">1245</FONT>         */<a name="line.1245"></a>
<FONT color="green">1246</FONT>        public static short sign(final short x) {<a name="line.1246"></a>
<FONT color="green">1247</FONT>            return (x == ZS) ? ZS : (x &gt; ZS) ? PS : NS;<a name="line.1247"></a>
<FONT color="green">1248</FONT>        }<a name="line.1248"></a>
<FONT color="green">1249</FONT>    <a name="line.1249"></a>
<FONT color="green">1250</FONT>        /**<a name="line.1250"></a>
<FONT color="green">1251</FONT>         * Returns the &lt;a href="http://mathworld.wolfram.com/HyperbolicSine.html"&gt;<a name="line.1251"></a>
<FONT color="green">1252</FONT>         * hyperbolic sine&lt;/a&gt; of x.<a name="line.1252"></a>
<FONT color="green">1253</FONT>         * <a name="line.1253"></a>
<FONT color="green">1254</FONT>         * @param x double value for which to find the hyperbolic sine<a name="line.1254"></a>
<FONT color="green">1255</FONT>         * @return hyperbolic sine of x<a name="line.1255"></a>
<FONT color="green">1256</FONT>         */<a name="line.1256"></a>
<FONT color="green">1257</FONT>        public static double sinh(double x) {<a name="line.1257"></a>
<FONT color="green">1258</FONT>            return (Math.exp(x) - Math.exp(-x)) / 2.0;<a name="line.1258"></a>
<FONT color="green">1259</FONT>        }<a name="line.1259"></a>
<FONT color="green">1260</FONT>    <a name="line.1260"></a>
<FONT color="green">1261</FONT>        /**<a name="line.1261"></a>
<FONT color="green">1262</FONT>         * Subtract two integers, checking for overflow.<a name="line.1262"></a>
<FONT color="green">1263</FONT>         * <a name="line.1263"></a>
<FONT color="green">1264</FONT>         * @param x the minuend<a name="line.1264"></a>
<FONT color="green">1265</FONT>         * @param y the subtrahend<a name="line.1265"></a>
<FONT color="green">1266</FONT>         * @return the difference &lt;code&gt;x-y&lt;/code&gt;<a name="line.1266"></a>
<FONT color="green">1267</FONT>         * @throws ArithmeticException if the result can not be represented as an<a name="line.1267"></a>
<FONT color="green">1268</FONT>         *         int<a name="line.1268"></a>
<FONT color="green">1269</FONT>         * @since 1.1<a name="line.1269"></a>
<FONT color="green">1270</FONT>         */<a name="line.1270"></a>
<FONT color="green">1271</FONT>        public static int subAndCheck(int x, int y) {<a name="line.1271"></a>
<FONT color="green">1272</FONT>            long s = (long)x - (long)y;<a name="line.1272"></a>
<FONT color="green">1273</FONT>            if (s &lt; Integer.MIN_VALUE || s &gt; Integer.MAX_VALUE) {<a name="line.1273"></a>
<FONT color="green">1274</FONT>                throw new ArithmeticException("overflow: subtract");<a name="line.1274"></a>
<FONT color="green">1275</FONT>            }<a name="line.1275"></a>
<FONT color="green">1276</FONT>            return (int)s;<a name="line.1276"></a>
<FONT color="green">1277</FONT>        }<a name="line.1277"></a>
<FONT color="green">1278</FONT>    <a name="line.1278"></a>
<FONT color="green">1279</FONT>        /**<a name="line.1279"></a>
<FONT color="green">1280</FONT>         * Subtract two long integers, checking for overflow.<a name="line.1280"></a>
<FONT color="green">1281</FONT>         * <a name="line.1281"></a>
<FONT color="green">1282</FONT>         * @param a first value<a name="line.1282"></a>
<FONT color="green">1283</FONT>         * @param b second value<a name="line.1283"></a>
<FONT color="green">1284</FONT>         * @return the difference &lt;code&gt;a-b&lt;/code&gt;<a name="line.1284"></a>
<FONT color="green">1285</FONT>         * @throws ArithmeticException if the result can not be represented as an<a name="line.1285"></a>
<FONT color="green">1286</FONT>         *         long<a name="line.1286"></a>
<FONT color="green">1287</FONT>         * @since 1.2<a name="line.1287"></a>
<FONT color="green">1288</FONT>         */<a name="line.1288"></a>
<FONT color="green">1289</FONT>        public static long subAndCheck(long a, long b) {<a name="line.1289"></a>
<FONT color="green">1290</FONT>            long ret;<a name="line.1290"></a>
<FONT color="green">1291</FONT>            String msg = "overflow: subtract";<a name="line.1291"></a>
<FONT color="green">1292</FONT>            if (b == Long.MIN_VALUE) {<a name="line.1292"></a>
<FONT color="green">1293</FONT>                if (a &lt; 0) {<a name="line.1293"></a>
<FONT color="green">1294</FONT>                    ret = a - b;<a name="line.1294"></a>
<FONT color="green">1295</FONT>                } else {<a name="line.1295"></a>
<FONT color="green">1296</FONT>                    throw new ArithmeticException(msg);<a name="line.1296"></a>
<FONT color="green">1297</FONT>                }<a name="line.1297"></a>
<FONT color="green">1298</FONT>            } else {<a name="line.1298"></a>
<FONT color="green">1299</FONT>                // use additive inverse<a name="line.1299"></a>
<FONT color="green">1300</FONT>                ret = addAndCheck(a, -b, msg);<a name="line.1300"></a>
<FONT color="green">1301</FONT>            }<a name="line.1301"></a>
<FONT color="green">1302</FONT>            return ret;<a name="line.1302"></a>
<FONT color="green">1303</FONT>        }<a name="line.1303"></a>
<FONT color="green">1304</FONT>    <a name="line.1304"></a>
<FONT color="green">1305</FONT>        /**<a name="line.1305"></a>
<FONT color="green">1306</FONT>         * Raise an int to an int power.<a name="line.1306"></a>
<FONT color="green">1307</FONT>         * @param k number to raise<a name="line.1307"></a>
<FONT color="green">1308</FONT>         * @param e exponent (must be positive or null)<a name="line.1308"></a>
<FONT color="green">1309</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.1309"></a>
<FONT color="green">1310</FONT>         * @exception IllegalArgumentException if e is negative<a name="line.1310"></a>
<FONT color="green">1311</FONT>         */<a name="line.1311"></a>
<FONT color="green">1312</FONT>        public static int pow(final int k, int e)<a name="line.1312"></a>
<FONT color="green">1313</FONT>            throws IllegalArgumentException {<a name="line.1313"></a>
<FONT color="green">1314</FONT>    <a name="line.1314"></a>
<FONT color="green">1315</FONT>            if (e &lt; 0) {<a name="line.1315"></a>
<FONT color="green">1316</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1316"></a>
<FONT color="green">1317</FONT>                    "cannot raise an integral value to a negative power ({0}^{1})",<a name="line.1317"></a>
<FONT color="green">1318</FONT>                    k, e);<a name="line.1318"></a>
<FONT color="green">1319</FONT>            }<a name="line.1319"></a>
<FONT color="green">1320</FONT>    <a name="line.1320"></a>
<FONT color="green">1321</FONT>            int result = 1;<a name="line.1321"></a>
<FONT color="green">1322</FONT>            int k2p    = k;<a name="line.1322"></a>
<FONT color="green">1323</FONT>            while (e != 0) {<a name="line.1323"></a>
<FONT color="green">1324</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.1324"></a>
<FONT color="green">1325</FONT>                    result *= k2p;<a name="line.1325"></a>
<FONT color="green">1326</FONT>                }<a name="line.1326"></a>
<FONT color="green">1327</FONT>                k2p *= k2p;<a name="line.1327"></a>
<FONT color="green">1328</FONT>                e = e &gt;&gt; 1;<a name="line.1328"></a>
<FONT color="green">1329</FONT>            }<a name="line.1329"></a>
<FONT color="green">1330</FONT>    <a name="line.1330"></a>
<FONT color="green">1331</FONT>            return result;<a name="line.1331"></a>
<FONT color="green">1332</FONT>    <a name="line.1332"></a>
<FONT color="green">1333</FONT>        }<a name="line.1333"></a>
<FONT color="green">1334</FONT>    <a name="line.1334"></a>
<FONT color="green">1335</FONT>        /**<a name="line.1335"></a>
<FONT color="green">1336</FONT>         * Raise an int to a long power.<a name="line.1336"></a>
<FONT color="green">1337</FONT>         * @param k number to raise<a name="line.1337"></a>
<FONT color="green">1338</FONT>         * @param e exponent (must be positive or null)<a name="line.1338"></a>
<FONT color="green">1339</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.1339"></a>
<FONT color="green">1340</FONT>         * @exception IllegalArgumentException if e is negative<a name="line.1340"></a>
<FONT color="green">1341</FONT>         */<a name="line.1341"></a>
<FONT color="green">1342</FONT>        public static int pow(final int k, long e)<a name="line.1342"></a>
<FONT color="green">1343</FONT>            throws IllegalArgumentException {<a name="line.1343"></a>
<FONT color="green">1344</FONT>    <a name="line.1344"></a>
<FONT color="green">1345</FONT>            if (e &lt; 0) {<a name="line.1345"></a>
<FONT color="green">1346</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1346"></a>
<FONT color="green">1347</FONT>                    "cannot raise an integral value to a negative power ({0}^{1})",<a name="line.1347"></a>
<FONT color="green">1348</FONT>                    k, e);<a name="line.1348"></a>
<FONT color="green">1349</FONT>            }<a name="line.1349"></a>
<FONT color="green">1350</FONT>    <a name="line.1350"></a>
<FONT color="green">1351</FONT>            int result = 1;<a name="line.1351"></a>
<FONT color="green">1352</FONT>            int k2p    = k;<a name="line.1352"></a>
<FONT color="green">1353</FONT>            while (e != 0) {<a name="line.1353"></a>
<FONT color="green">1354</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.1354"></a>
<FONT color="green">1355</FONT>                    result *= k2p;<a name="line.1355"></a>
<FONT color="green">1356</FONT>                }<a name="line.1356"></a>
<FONT color="green">1357</FONT>                k2p *= k2p;<a name="line.1357"></a>
<FONT color="green">1358</FONT>                e = e &gt;&gt; 1;<a name="line.1358"></a>
<FONT color="green">1359</FONT>            }<a name="line.1359"></a>
<FONT color="green">1360</FONT>    <a name="line.1360"></a>
<FONT color="green">1361</FONT>            return result;<a name="line.1361"></a>
<FONT color="green">1362</FONT>    <a name="line.1362"></a>
<FONT color="green">1363</FONT>        }<a name="line.1363"></a>
<FONT color="green">1364</FONT>    <a name="line.1364"></a>
<FONT color="green">1365</FONT>        /**<a name="line.1365"></a>
<FONT color="green">1366</FONT>         * Raise a long to an int power.<a name="line.1366"></a>
<FONT color="green">1367</FONT>         * @param k number to raise<a name="line.1367"></a>
<FONT color="green">1368</FONT>         * @param e exponent (must be positive or null)<a name="line.1368"></a>
<FONT color="green">1369</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.1369"></a>
<FONT color="green">1370</FONT>         * @exception IllegalArgumentException if e is negative<a name="line.1370"></a>
<FONT color="green">1371</FONT>         */<a name="line.1371"></a>
<FONT color="green">1372</FONT>        public static long pow(final long k, int e)<a name="line.1372"></a>
<FONT color="green">1373</FONT>            throws IllegalArgumentException {<a name="line.1373"></a>
<FONT color="green">1374</FONT>    <a name="line.1374"></a>
<FONT color="green">1375</FONT>            if (e &lt; 0) {<a name="line.1375"></a>
<FONT color="green">1376</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1376"></a>
<FONT color="green">1377</FONT>                    "cannot raise an integral value to a negative power ({0}^{1})",<a name="line.1377"></a>
<FONT color="green">1378</FONT>                    k, e);<a name="line.1378"></a>
<FONT color="green">1379</FONT>            }<a name="line.1379"></a>
<FONT color="green">1380</FONT>    <a name="line.1380"></a>
<FONT color="green">1381</FONT>            long result = 1l;<a name="line.1381"></a>
<FONT color="green">1382</FONT>            long k2p    = k;<a name="line.1382"></a>
<FONT color="green">1383</FONT>            while (e != 0) {<a name="line.1383"></a>
<FONT color="green">1384</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.1384"></a>
<FONT color="green">1385</FONT>                    result *= k2p;<a name="line.1385"></a>
<FONT color="green">1386</FONT>                }<a name="line.1386"></a>
<FONT color="green">1387</FONT>                k2p *= k2p;<a name="line.1387"></a>
<FONT color="green">1388</FONT>                e = e &gt;&gt; 1;<a name="line.1388"></a>
<FONT color="green">1389</FONT>            }<a name="line.1389"></a>
<FONT color="green">1390</FONT>    <a name="line.1390"></a>
<FONT color="green">1391</FONT>            return result;<a name="line.1391"></a>
<FONT color="green">1392</FONT>    <a name="line.1392"></a>
<FONT color="green">1393</FONT>        }<a name="line.1393"></a>
<FONT color="green">1394</FONT>    <a name="line.1394"></a>
<FONT color="green">1395</FONT>        /**<a name="line.1395"></a>
<FONT color="green">1396</FONT>         * Raise a long to a long power.<a name="line.1396"></a>
<FONT color="green">1397</FONT>         * @param k number to raise<a name="line.1397"></a>
<FONT color="green">1398</FONT>         * @param e exponent (must be positive or null)<a name="line.1398"></a>
<FONT color="green">1399</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.1399"></a>
<FONT color="green">1400</FONT>         * @exception IllegalArgumentException if e is negative<a name="line.1400"></a>
<FONT color="green">1401</FONT>         */<a name="line.1401"></a>
<FONT color="green">1402</FONT>        public static long pow(final long k, long e)<a name="line.1402"></a>
<FONT color="green">1403</FONT>            throws IllegalArgumentException {<a name="line.1403"></a>
<FONT color="green">1404</FONT>    <a name="line.1404"></a>
<FONT color="green">1405</FONT>            if (e &lt; 0) {<a name="line.1405"></a>
<FONT color="green">1406</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1406"></a>
<FONT color="green">1407</FONT>                    "cannot raise an integral value to a negative power ({0}^{1})",<a name="line.1407"></a>
<FONT color="green">1408</FONT>                    k, e);<a name="line.1408"></a>
<FONT color="green">1409</FONT>            }<a name="line.1409"></a>
<FONT color="green">1410</FONT>    <a name="line.1410"></a>
<FONT color="green">1411</FONT>            long result = 1l;<a name="line.1411"></a>
<FONT color="green">1412</FONT>            long k2p    = k;<a name="line.1412"></a>
<FONT color="green">1413</FONT>            while (e != 0) {<a name="line.1413"></a>
<FONT color="green">1414</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.1414"></a>
<FONT color="green">1415</FONT>                    result *= k2p;<a name="line.1415"></a>
<FONT color="green">1416</FONT>                }<a name="line.1416"></a>
<FONT color="green">1417</FONT>                k2p *= k2p;<a name="line.1417"></a>
<FONT color="green">1418</FONT>                e = e &gt;&gt; 1;<a name="line.1418"></a>
<FONT color="green">1419</FONT>            }<a name="line.1419"></a>
<FONT color="green">1420</FONT>    <a name="line.1420"></a>
<FONT color="green">1421</FONT>            return result;<a name="line.1421"></a>
<FONT color="green">1422</FONT>    <a name="line.1422"></a>
<FONT color="green">1423</FONT>        }<a name="line.1423"></a>
<FONT color="green">1424</FONT>    <a name="line.1424"></a>
<FONT color="green">1425</FONT>        /**<a name="line.1425"></a>
<FONT color="green">1426</FONT>         * Raise a BigInteger to an int power.<a name="line.1426"></a>
<FONT color="green">1427</FONT>         * @param k number to raise<a name="line.1427"></a>
<FONT color="green">1428</FONT>         * @param e exponent (must be positive or null)<a name="line.1428"></a>
<FONT color="green">1429</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.1429"></a>
<FONT color="green">1430</FONT>         * @exception IllegalArgumentException if e is negative<a name="line.1430"></a>
<FONT color="green">1431</FONT>         */<a name="line.1431"></a>
<FONT color="green">1432</FONT>        public static BigInteger pow(final BigInteger k, int e)<a name="line.1432"></a>
<FONT color="green">1433</FONT>            throws IllegalArgumentException {<a name="line.1433"></a>
<FONT color="green">1434</FONT>    <a name="line.1434"></a>
<FONT color="green">1435</FONT>            if (e &lt; 0) {<a name="line.1435"></a>
<FONT color="green">1436</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1436"></a>
<FONT color="green">1437</FONT>                    "cannot raise an integral value to a negative power ({0}^{1})",<a name="line.1437"></a>
<FONT color="green">1438</FONT>                    k, e);<a name="line.1438"></a>
<FONT color="green">1439</FONT>            }<a name="line.1439"></a>
<FONT color="green">1440</FONT>    <a name="line.1440"></a>
<FONT color="green">1441</FONT>            return k.pow(e);<a name="line.1441"></a>
<FONT color="green">1442</FONT>    <a name="line.1442"></a>
<FONT color="green">1443</FONT>        }<a name="line.1443"></a>
<FONT color="green">1444</FONT>    <a name="line.1444"></a>
<FONT color="green">1445</FONT>        /**<a name="line.1445"></a>
<FONT color="green">1446</FONT>         * Raise a BigInteger to a long power.<a name="line.1446"></a>
<FONT color="green">1447</FONT>         * @param k number to raise<a name="line.1447"></a>
<FONT color="green">1448</FONT>         * @param e exponent (must be positive or null)<a name="line.1448"></a>
<FONT color="green">1449</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.1449"></a>
<FONT color="green">1450</FONT>         * @exception IllegalArgumentException if e is negative<a name="line.1450"></a>
<FONT color="green">1451</FONT>         */<a name="line.1451"></a>
<FONT color="green">1452</FONT>        public static BigInteger pow(final BigInteger k, long e)<a name="line.1452"></a>
<FONT color="green">1453</FONT>            throws IllegalArgumentException {<a name="line.1453"></a>
<FONT color="green">1454</FONT>    <a name="line.1454"></a>
<FONT color="green">1455</FONT>            if (e &lt; 0) {<a name="line.1455"></a>
<FONT color="green">1456</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1456"></a>
<FONT color="green">1457</FONT>                    "cannot raise an integral value to a negative power ({0}^{1})",<a name="line.1457"></a>
<FONT color="green">1458</FONT>                    k, e);<a name="line.1458"></a>
<FONT color="green">1459</FONT>            }<a name="line.1459"></a>
<FONT color="green">1460</FONT>    <a name="line.1460"></a>
<FONT color="green">1461</FONT>            BigInteger result = BigInteger.ONE;<a name="line.1461"></a>
<FONT color="green">1462</FONT>            BigInteger k2p    = k;<a name="line.1462"></a>
<FONT color="green">1463</FONT>            while (e != 0) {<a name="line.1463"></a>
<FONT color="green">1464</FONT>                if ((e &amp; 0x1) != 0) {<a name="line.1464"></a>
<FONT color="green">1465</FONT>                    result = result.multiply(k2p);<a name="line.1465"></a>
<FONT color="green">1466</FONT>                }<a name="line.1466"></a>
<FONT color="green">1467</FONT>                k2p = k2p.multiply(k2p);<a name="line.1467"></a>
<FONT color="green">1468</FONT>                e = e &gt;&gt; 1;<a name="line.1468"></a>
<FONT color="green">1469</FONT>            }<a name="line.1469"></a>
<FONT color="green">1470</FONT>    <a name="line.1470"></a>
<FONT color="green">1471</FONT>            return result;<a name="line.1471"></a>
<FONT color="green">1472</FONT>    <a name="line.1472"></a>
<FONT color="green">1473</FONT>        }<a name="line.1473"></a>
<FONT color="green">1474</FONT>    <a name="line.1474"></a>
<FONT color="green">1475</FONT>        /**<a name="line.1475"></a>
<FONT color="green">1476</FONT>         * Raise a BigInteger to a BigInteger power.<a name="line.1476"></a>
<FONT color="green">1477</FONT>         * @param k number to raise<a name="line.1477"></a>
<FONT color="green">1478</FONT>         * @param e exponent (must be positive or null)<a name="line.1478"></a>
<FONT color="green">1479</FONT>         * @return k&lt;sup&gt;e&lt;/sup&gt;<a name="line.1479"></a>
<FONT color="green">1480</FONT>         * @exception IllegalArgumentException if e is negative<a name="line.1480"></a>
<FONT color="green">1481</FONT>         */<a name="line.1481"></a>
<FONT color="green">1482</FONT>        public static BigInteger pow(final BigInteger k, BigInteger e)<a name="line.1482"></a>
<FONT color="green">1483</FONT>            throws IllegalArgumentException {<a name="line.1483"></a>
<FONT color="green">1484</FONT>    <a name="line.1484"></a>
<FONT color="green">1485</FONT>            if (e.compareTo(BigInteger.ZERO) &lt; 0) {<a name="line.1485"></a>
<FONT color="green">1486</FONT>                throw MathRuntimeException.createIllegalArgumentException(<a name="line.1486"></a>
<FONT color="green">1487</FONT>                    "cannot raise an integral value to a negative power ({0}^{1})",<a name="line.1487"></a>
<FONT color="green">1488</FONT>                    k, e);<a name="line.1488"></a>
<FONT color="green">1489</FONT>            }<a name="line.1489"></a>
<FONT color="green">1490</FONT>    <a name="line.1490"></a>
<FONT color="green">1491</FONT>            BigInteger result = BigInteger.ONE;<a name="line.1491"></a>
<FONT color="green">1492</FONT>            BigInteger k2p    = k;<a name="line.1492"></a>
<FONT color="green">1493</FONT>            while (!BigInteger.ZERO.equals(e)) {<a name="line.1493"></a>
<FONT color="green">1494</FONT>                if (e.testBit(0)) {<a name="line.1494"></a>
<FONT color="green">1495</FONT>                    result = result.multiply(k2p);<a name="line.1495"></a>
<FONT color="green">1496</FONT>                }<a name="line.1496"></a>
<FONT color="green">1497</FONT>                k2p = k2p.multiply(k2p);<a name="line.1497"></a>
<FONT color="green">1498</FONT>                e = e.shiftRight(1);<a name="line.1498"></a>
<FONT color="green">1499</FONT>            }<a name="line.1499"></a>
<FONT color="green">1500</FONT>    <a name="line.1500"></a>
<FONT color="green">1501</FONT>            return result;<a name="line.1501"></a>
<FONT color="green">1502</FONT>    <a name="line.1502"></a>
<FONT color="green">1503</FONT>        }<a name="line.1503"></a>
<FONT color="green">1504</FONT>    <a name="line.1504"></a>
<FONT color="green">1505</FONT>        /**<a name="line.1505"></a>
<FONT color="green">1506</FONT>         * Calculates the L&lt;sub&gt;1&lt;/sub&gt; (sum of abs) distance between two points.<a name="line.1506"></a>
<FONT color="green">1507</FONT>         *<a name="line.1507"></a>
<FONT color="green">1508</FONT>         * @param p1 the first point<a name="line.1508"></a>
<FONT color="green">1509</FONT>         * @param p2 the second point<a name="line.1509"></a>
<FONT color="green">1510</FONT>         * @return the L&lt;sub&gt;1&lt;/sub&gt; distance between the two points<a name="line.1510"></a>
<FONT color="green">1511</FONT>         */<a name="line.1511"></a>
<FONT color="green">1512</FONT>        public static final double distance1(double[] p1, double[] p2) {<a name="line.1512"></a>
<FONT color="green">1513</FONT>            double sum = 0;<a name="line.1513"></a>
<FONT color="green">1514</FONT>            for (int i = 0; i &lt; p1.length; i++) {<a name="line.1514"></a>
<FONT color="green">1515</FONT>                sum += Math.abs(p1[i] - p2[i]);<a name="line.1515"></a>
<FONT color="green">1516</FONT>            }<a name="line.1516"></a>
<FONT color="green">1517</FONT>            return sum;<a name="line.1517"></a>
<FONT color="green">1518</FONT>        }<a name="line.1518"></a>
<FONT color="green">1519</FONT>        <a name="line.1519"></a>
<FONT color="green">1520</FONT>        /**<a name="line.1520"></a>
<FONT color="green">1521</FONT>         * Calculates the L&lt;sub&gt;1&lt;/sub&gt; (sum of abs) distance between two points.<a name="line.1521"></a>
<FONT color="green">1522</FONT>         *<a name="line.1522"></a>
<FONT color="green">1523</FONT>         * @param p1 the first point<a name="line.1523"></a>
<FONT color="green">1524</FONT>         * @param p2 the second point<a name="line.1524"></a>
<FONT color="green">1525</FONT>         * @return the L&lt;sub&gt;1&lt;/sub&gt; distance between the two points<a name="line.1525"></a>
<FONT color="green">1526</FONT>         */<a name="line.1526"></a>
<FONT color="green">1527</FONT>        public static final int distance1(int[] p1, int[] p2) {<a name="line.1527"></a>
<FONT color="green">1528</FONT>          int sum = 0;<a name="line.1528"></a>
<FONT color="green">1529</FONT>          for (int i = 0; i &lt; p1.length; i++) {<a name="line.1529"></a>
<FONT color="green">1530</FONT>              sum += Math.abs(p1[i] - p2[i]);<a name="line.1530"></a>
<FONT color="green">1531</FONT>          }<a name="line.1531"></a>
<FONT color="green">1532</FONT>          return sum;<a name="line.1532"></a>
<FONT color="green">1533</FONT>        }<a name="line.1533"></a>
<FONT color="green">1534</FONT>    <a name="line.1534"></a>
<FONT color="green">1535</FONT>        /**<a name="line.1535"></a>
<FONT color="green">1536</FONT>         * Calculates the L&lt;sub&gt;2&lt;/sub&gt; (Euclidean) distance between two points.<a name="line.1536"></a>
<FONT color="green">1537</FONT>         *<a name="line.1537"></a>
<FONT color="green">1538</FONT>         * @param p1 the first point<a name="line.1538"></a>
<FONT color="green">1539</FONT>         * @param p2 the second point<a name="line.1539"></a>
<FONT color="green">1540</FONT>         * @return the L&lt;sub&gt;2&lt;/sub&gt; distance between the two points<a name="line.1540"></a>
<FONT color="green">1541</FONT>         */<a name="line.1541"></a>
<FONT color="green">1542</FONT>        public static final double distance(double[] p1, double[] p2) {<a name="line.1542"></a>
<FONT color="green">1543</FONT>            double sum = 0;<a name="line.1543"></a>
<FONT color="green">1544</FONT>            for (int i = 0; i &lt; p1.length; i++) {<a name="line.1544"></a>
<FONT color="green">1545</FONT>                final double dp = p1[i] - p2[i];<a name="line.1545"></a>
<FONT color="green">1546</FONT>                sum += dp * dp;<a name="line.1546"></a>
<FONT color="green">1547</FONT>            }<a name="line.1547"></a>
<FONT color="green">1548</FONT>            return Math.sqrt(sum);<a name="line.1548"></a>
<FONT color="green">1549</FONT>        }<a name="line.1549"></a>
<FONT color="green">1550</FONT>        <a name="line.1550"></a>
<FONT color="green">1551</FONT>        /**<a name="line.1551"></a>
<FONT color="green">1552</FONT>         * Calculates the L&lt;sub&gt;2&lt;/sub&gt; (Euclidean) distance between two points.<a name="line.1552"></a>
<FONT color="green">1553</FONT>         *<a name="line.1553"></a>
<FONT color="green">1554</FONT>         * @param p1 the first point<a name="line.1554"></a>
<FONT color="green">1555</FONT>         * @param p2 the second point<a name="line.1555"></a>
<FONT color="green">1556</FONT>         * @return the L&lt;sub&gt;2&lt;/sub&gt; distance between the two points<a name="line.1556"></a>
<FONT color="green">1557</FONT>         */<a name="line.1557"></a>
<FONT color="green">1558</FONT>        public static final double distance(int[] p1, int[] p2) {<a name="line.1558"></a>
<FONT color="green">1559</FONT>          int sum = 0;<a name="line.1559"></a>
<FONT color="green">1560</FONT>          for (int i = 0; i &lt; p1.length; i++) {<a name="line.1560"></a>
<FONT color="green">1561</FONT>              final int dp = p1[i] - p2[i];<a name="line.1561"></a>
<FONT color="green">1562</FONT>              sum += dp * dp;<a name="line.1562"></a>
<FONT color="green">1563</FONT>          }<a name="line.1563"></a>
<FONT color="green">1564</FONT>          return Math.sqrt(sum);<a name="line.1564"></a>
<FONT color="green">1565</FONT>        }<a name="line.1565"></a>
<FONT color="green">1566</FONT>        <a name="line.1566"></a>
<FONT color="green">1567</FONT>        /**<a name="line.1567"></a>
<FONT color="green">1568</FONT>         * Calculates the L&lt;sub&gt;&amp;infin;&lt;/sub&gt; (max of abs) distance between two points.<a name="line.1568"></a>
<FONT color="green">1569</FONT>         *<a name="line.1569"></a>
<FONT color="green">1570</FONT>         * @param p1 the first point<a name="line.1570"></a>
<FONT color="green">1571</FONT>         * @param p2 the second point<a name="line.1571"></a>
<FONT color="green">1572</FONT>         * @return the L&lt;sub&gt;&amp;infin;&lt;/sub&gt; distance between the two points<a name="line.1572"></a>
<FONT color="green">1573</FONT>         */<a name="line.1573"></a>
<FONT color="green">1574</FONT>        public static final double distanceInf(double[] p1, double[] p2) {<a name="line.1574"></a>
<FONT color="green">1575</FONT>            double max = 0;<a name="line.1575"></a>
<FONT color="green">1576</FONT>            for (int i = 0; i &lt; p1.length; i++) {<a name="line.1576"></a>
<FONT color="green">1577</FONT>                max = Math.max(max, Math.abs(p1[i] - p2[i]));<a name="line.1577"></a>
<FONT color="green">1578</FONT>            }<a name="line.1578"></a>
<FONT color="green">1579</FONT>            return max;<a name="line.1579"></a>
<FONT color="green">1580</FONT>        }<a name="line.1580"></a>
<FONT color="green">1581</FONT>        <a name="line.1581"></a>
<FONT color="green">1582</FONT>        /**<a name="line.1582"></a>
<FONT color="green">1583</FONT>         * Calculates the L&lt;sub&gt;&amp;infin;&lt;/sub&gt; (max of abs) distance between two points.<a name="line.1583"></a>
<FONT color="green">1584</FONT>         *<a name="line.1584"></a>
<FONT color="green">1585</FONT>         * @param p1 the first point<a name="line.1585"></a>
<FONT color="green">1586</FONT>         * @param p2 the second point<a name="line.1586"></a>
<FONT color="green">1587</FONT>         * @return the L&lt;sub&gt;&amp;infin;&lt;/sub&gt; distance between the two points<a name="line.1587"></a>
<FONT color="green">1588</FONT>         */<a name="line.1588"></a>
<FONT color="green">1589</FONT>        public static final int distanceInf(int[] p1, int[] p2) {<a name="line.1589"></a>
<FONT color="green">1590</FONT>            int max = 0;<a name="line.1590"></a>
<FONT color="green">1591</FONT>            for (int i = 0; i &lt; p1.length; i++) {<a name="line.1591"></a>
<FONT color="green">1592</FONT>                max = Math.max(max, Math.abs(p1[i] - p2[i]));<a name="line.1592"></a>
<FONT color="green">1593</FONT>            }<a name="line.1593"></a>
<FONT color="green">1594</FONT>            return max;<a name="line.1594"></a>
<FONT color="green">1595</FONT>        }<a name="line.1595"></a>
<FONT color="green">1596</FONT>    <a name="line.1596"></a>
<FONT color="green">1597</FONT>        <a name="line.1597"></a>
<FONT color="green">1598</FONT>    }<a name="line.1598"></a>




























































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